### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)



Le 30-mai-06, à 19:13, Tom Caylor wrote :

From what you've said about dovetailing before, you don't have to have
just a single sequence in order to dovetail.  You can jump among
multiple sequences.  I have yet to understand how you could dovetail on
something that is not effective.  That seems to be where you're going.
I guess we have to get to non-effective diagonalization first.

It is true that we can dovetail on multiple sequences, once we can
generate their codes, like all the sequence of growing computable
functions we have visited.
But none of them are universal.
Once I can name a sequence of growing function, and thus can program an
enumeration of their  codes, I will be able to diagonalize it, showing
it was not universal.

---

Quentin Anciaux wrote:

I think dovetailing is possible because the dovetailer only complete
sequences
at infinity. So when you construct the matrice on which you
will diagonalize, you are already diagonilizing it at the same time.
Example: when you have the first number of the first growing function,
you
can also have the first number of the diagonalize function (by adding
1) and
the first number of the diagonalize*diagonalize function and ... ad
recursum.
By dovetailing you execute in fact everything in parallel but all
infinites
sequences are only completed at infinity.

Same answer as the one for Tom. You can diagonalize only on the
transfinite sequences up to a nameable ordinal, and clearly this cannot
be closed for diagonalization. Even in the limit, the transfinite
construction will fail to name some computable growing function.

---

Hal Finney wrote:

The dovetailer I know does not seem relevant to this discussion about
functions.  It generates programs, not functions.

So does our sequence of growing functions. They are given by the
programmable generation of the code of the growing function. The same
for the diagonalization.

For example, it
generates all 1 bit programs and runs each for one cycle; then
generates
all 2 bit programs and runs each for 2 cycles; then generates all 3
bit programs and runs each for 3 cycles; and so on indefinitely.  (This
assumes that the 3 bit programs include all 2- and 1-bit programs,
etc.)
In this way all programs get run with an arbitrary number of cycles.

Close :)

-
Quentin Anciaux comments on Hal Finney:

In fact it is relevant because of this :
- Bruno showed us that it is not possible to write a program that will
list
sequentially all growing functions.

...that will list sequentially all computable growing functions. Right.

- But the dovetailer will not do it too, but what it will do instead is
generate all program that list all growing functions.

Mmh...

So the dovetailer will not list all the growing function but
will generate (and execute in dovetailing) the infinite sequence of
programs
that generate all of them.

The shadow of the truth, but what you describe would still be
diagonalized out. (Or you are very unclear, to be sure, you will tell
me later ... or you will not tell me later :)

--

Jesse Mazer (on Hal Finney):

I was being a little sloppy...it's true that a non-halting program
would not
be equivalent to a computable function, but I think we can at least
say that
the set of all computable functions is a *subset* of the set of all
programs.

Key remark.

As for the programs taking input or not, if you look at the set of
all programs operating on finite input strings, each one of these can
be
emulated by a program which has no input string (the input string is
built
into the design of the program).

Actually this is a key remark too, but it will be needed only later (I
recall that I am trying to explain a the missing link between computer
science and Smullyan's heart of the Matter in FU, after a question by
George Levy). This key remark is related to a simple but basic theorem
in computer science known as the SMN theorem, S for substitution and M
and N are parameter, and grosso modo the theorem says that you can
programs can be mechanically parametrized by freezing some of their
inputs.

So for any computable function, there
should be some member of the set of all halting programs being run by
the
dovetailer that gives the same output as the function, no?

Yes. Do you see or know or guess the many consequences?

---

George Levy wrote:

To speak only for myself,  I think I have a sufficient understanding of
the thread. Essentially you have shown that one cannot form a set of
all
numbers/functions because given any set of numbers/functions it is
always possible, using diagonalization,  to generate new
numbers/functions: the Plenitude is too large to be a set. This leads
to
a problem with the assumption of the 

### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


Le Mercredi 31 Mai 2006 00:21, Hal Finney a écrit :
The dovetailer I know does not seem relevant to this discussion about
functions.  It generates programs, not functions.  For example, it
generates all 1 bit programs and runs each for one cycle; then generates
all 2 bit programs and runs each for 2 cycles; then generates all 3
bit programs and runs each for 3 cycles; and so on indefinitely.  (This
assumes that the 3 bit programs include all 2- and 1-bit programs, etc.)
In this way all programs get run with an arbitrary number of cycles.

In fact it is relevant because of this :
- Bruno showed us that it is not possible to write a program that will list
sequentially all growing functions.
- But the dovetailer will not do it too, but what it will do instead is
generate all program that list all growing functions.

So it will first generate the programme that create the first sequence and
also the pogram that create the sequence composed of diagonalisation of the
first and so on... it can because program are countable because they are
mapped to N. So the dovetailer will not list all the growing function but
will generate (and execute in dovetailing) the infinite sequence of programs
that generate all of them.

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### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


Hal Finney wrote:

Jesse Mazer writes:
The dovetailer is only supposed to generate all *computable* functions
though, correct? And the diagonalization of the (countable) set of all
computable functions would not itself be computable.

The dovetailer I know does not seem relevant to this discussion about
functions.  It generates programs, not functions.  For example, it
generates all 1 bit programs and runs each for one cycle; then generates
all 2 bit programs and runs each for 2 cycles; then generates all 3
bit programs and runs each for 3 cycles; and so on indefinitely.  (This
assumes that the 3 bit programs include all 2- and 1-bit programs, etc.)
In this way all programs get run with an arbitrary number of cycles.

These programs differ from functions in two ways.  First, programs may
never halt and hence may produce no fixed output, while functions must
have a well defined result.  And second, these programs take no inputs,
while functions should have at least one input variable.

What do you understand a dovetailer to be, in the context of computable
functions?

Hal Finney

I was being a little sloppy...it's true that a non-halting program would not
be equivalent to a computable function, but I think we can at least say that
the set of all computable functions is a *subset* of the set of all
programs. As for the programs taking input or not, if you look at the set of
all programs operating on finite input strings, each one of these can be
emulated by a program which has no input string (the input string is built
into the design of the program). So for any computable function, there
should be some member of the set of all halting programs being run by the
dovetailer that gives the same output as the function, no?

Jesse

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### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)



Le 30-mai-06, à 03:14, Tom Caylor a écrit :

OK.  I see that so far (above) there's no problem.  (See below for
where I still have concern(s).)  Here I was taking a fixed N, but G is
defined as the diagonal, so my comparison is not valid, and so my proof
that G is infinite for a fixed N is not valid.  I was taking G's
assignment of an ordinal of omega as being that it is in every way
larger than all Fi's, but in fact G is larger than all Fi's only when
comparing G(n) to Fn(n), not when comparing G(Nfixed) to Fi(N) for all
i's.

Right.

OK.  I think you are just throwing me off with your notation.  Do you
have to use transfinite ordinals (omega) to do this?  Couldn't you just
stay with successively combining and diagonalizing, like below, without
using omegas?

G(n) = Fn(n)+1
Gi(n) = G(...G(n)), G taken i times
subscripts...
H1(n) = Gn(n)+1
H1i(n) = H1(...H1(n)), H1 taken i times
H2(n) = H1n(n)+1
H2i(n) = H2(...H2(n)), H2 taken i times

Then the subscripts count the number of diagonalizations you've done,
and every time you do the Ackermann thing, instead of adding an omega
do the Ackermann thing with the *number* of subscripts, i.e. do the
Ackermann thing on the number of times you've done the Ackermann
thing... etc.

This may just be a technical point, but it doesn't seem precise to do
very much arithmetic with ordinals, like doing omega [omega] omega,
because you're just ordering things, and after a while you forget the
computations that are being performed.  I can see that it works for
just proving that you can continue to diagonalize and grow, which is
what you're doing.  I just don't want to be caught off guard and
suddenly realize you've slipped actual infinities in without me
realizing it.  I don't think you have.

OK. And you are right, I could have done this without mentioning the
constructive ordinal. But it is worth mentioning it, even at this early
stages, because they will reappear again and again.
Note that all those infinite but constructive ordinal are all countable
(in bijection with N), and even constructively so.  Note also, if you
haven't already done so, that omega is just N, the set of natural
numbers. I will soon give a more set-theoretical motivation for those
ordinals.

Actually there is a cute theorem about constructive ordinal. Indeed
they are equivalent to the recursive (programmable) linear
well-ordering on the natural numbers. Examples:

An order of type omega: the usual order on N (0123456...)

An order of type omega+1 : just decide that 0 is bigger than any non
null natural numbers:

123456  0

It is recursive in the sense that you can write a program FORTRAN (say)
capable of deciding it. For example such a program would stop on yes

An order of type omega+omega: just decide that all odd numbers are
bigger than the even ones, and take the usual order in case the two
numbers which are compared have the same parity:

0246810 . 13579...

An order of type omega+omega+omega: just decide that multiple of 3 are
bigger than the multiple of two, themselves bigger than the remaining
numbers:

15711131417...  0246810... 3691215...

Again it should be easy to write a Fortran program capable of deciding
that order (that is to decide for any x and y if x  y with that
(unusual) order.

Exercise: could you find an order of type omega*omega? (Hint: use the
prime numbers).

Those omega names are quite standard.

OK.  So we haven't left the finite behind yet.  It makes intuitive
sense to me that you can diagonalize till the cows come home, staying
within countability, and still not be done.  Otherwise infinity
wouldn't be infinite.

On the tricky question, it also makes intuitive sense that you can
sequence effectively on all computable growing functions.  This is
because the larger the growing function gets, the more uncovered space
(gaps) there are between the computable functions.  Any scheme for
generating growing functions will also leave behind every-growing
uncomputed gaps.  Very unmathematical of me to be so vague, but you've
already given us the answer, and I know you will fill in the gaps.  :)

I will. Unfortunately this week is full of duty charges. Meanwhile, I
would like to ask George and the others if they have a good
understanding of the present thread, that is on the fact that growing
functions has been well defined, that each sequence of such functions
are well defined, and each diagonalisation defines quite well a precise
programmable growing function (growing faster than the one in the
sequence it comes from).
Just a tiny effort, and I think we will have all we need to go into the
heart of the matter, and to understand why comp makes our universe
a godelized one in the Smullyan sense.

I meant 

### Re: Smullyan Shmullyan, give me a real example


Bruno,

It's been a long holiday weekend here in the US, Bruno,
responce.

Fromconventional math, everything you said was
correct, put to me by a co-list friend as .. should I

1m$that is: 0m$
:-) .

Well, I'm not sending you 1m$, but I will continue commentary. Consider for a moment, the possibility that the entire used ediface of mathematics is an analog of Abbott's Flatland. That though we may think we are 'calculating' in a completely identified domain, that the 'environment' of mathematics is extensive in new ways, and that there are new/different operators needed to access the extended mathematics. Consider G.Cantor. Suppose I said that not only are Aleph0 regions of math calculations, but that addional functions make all of those infinities - calculation accessible. That 'normal math' still applies .. but if and only if .. notated as referencing each frame-of-reference Aleph n. That to segue (equationally transduce) from any Aleph to any Aleph requires additional notations marks, in order to keep separate what Aleph the immediate notation referneces, or, mores into or out of. You remarked that it is absurd to : From (-5)^2 = 5^2 you will not infer that 5 = (-5), right? Actually, what I suggest does -relate- to this question. We make such presumption about positive or absolute value numeration that when we do back-functioning we overlook relations and information that might be inconvenient or cumbersome to treat. Such as differentiating an already integrated operation. That pesky throw-away scalar transform value of (+C) is unceremoniously thrown out because we assume is to be a non-consequential shift- or spatial-translation factor that needn't be considered in mathematical generalization. When we take a square-root, we ignore the minus signs option. When we look at quantum equations, we keep the positive set and ignore the negative set .. which in and of itself is contrary to quantum-math philosophy .. where all variables are included, even if anti-thetical. [M and not-M are concurrent rather than computationally mutually exclusive.] A closing thought for this morning (possible discussion of particulars being left for another day): --from an off-list letter, same list-subject Dear __ , I am broaching a substantially new logic. 1m$ that is: 0m\$

-is- a patent absurdity in current math.

The version that I came up with essentially
restructures the analysis of mathematics
as comparisons of dimensions.  I did one analysis
around the pythagorean theorem that results
in a statement  b=b^2 for any and all numbers, b.
[with the autonomous inclusion of new +/- markers
that arrive everytime a dimension is added to
or calculated to.]

What is missing in math notation are markers that
help a person to remember they may be co-navigating
several different dimensional fields at the same time,
where the left side of an equation is in 'm' dimensions
and the right in 'other than m' dimensions, yet
the equation is valid.  The trouble persists if the
notations presume that native dimensionality on both
sides is identical.

In -that- presumption, the numbers have to match conventional
math concepts and no such thing as  b=b^2 for any and all
numbers, b is allowed or even sensible.

It is like trying to have perfect translation
among human languages.  Not possible.  It's only
when we convert languages into the larger
information network of memes, that 'equal translation'
makes sense.  That's what I'm doing.  Identifying
a core realm of 'information' (albeit, mathematical
notions, concepts, information) that can transduce
as real and valid 'equalities' across the equals sign.

When the realm of dimensions is recognized as the
larger realm of mathematical memes.

If a person doesn't do that shift of
consciousness/sensibilities, they'll never 'get it'.

... but I see a shining country of mathematics that
no one else seems to recognize .. yet.   Jamie

Bruno, I know you are still going to treat this line of
thought/conversation as sophomoric. A natural reaction.
I can assure you it is 'of significance' however.

Best Regards,
Jamie Rose

Bruno Marchal wrote:

Le 26-mai-06, à 02:50, James N Rose a écrit :

An example at the core of it is a most simplistic
definition/equation.

1^1 = 1^0

[one to the exponent one  equals  one to the exponent zero]

To all mathematicians, this is a toss-out absurdity, with
no 'real meaning'.  n^0 is a convenience tool at best ;

n^0 = 1, because 1= (n^m)/(n^m) = n^(m-m) = n^0.
Or better n^0 = the number of functions from the empty
set (cardinal 0) to the set with cardinal n. This
justifies also 0^0 = 1 (there is one (empty)
function from the empty set to the empty set).

along with  'n/0 is 'undefined''.   We note the consistent/valid
notation, but walk away from any 

### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


Bruno Marchal wrote:

OK. And you are right, I could have done this without mentioning the
constructive ordinal. But it is worth mentioning it, even at this early
stages, because they will reappear again and again.
Note that all those infinite but constructive ordinal are all countable
(in bijection with N), and even constructively so.  Note also, if you
haven't already done so, that omega is just N, the set of natural
numbers. I will soon give a more set-theoretical motivation for those
ordinals.

OK.  I feel power sets coming.

Actually there is a cute theorem about constructive ordinal. Indeed
they are equivalent to the recursive (programmable) linear
well-ordering on the natural numbers. Examples:

An order of type omega: the usual order on N (0123456...)

An order of type omega+1 : just decide that 0 is bigger than any non
null natural numbers:

123456  0

It is recursive in the sense that you can write a program FORTRAN (say)
capable of deciding it. For example such a program would stop on yes

An order of type omega+omega: just decide that all odd numbers are
bigger than the even ones, and take the usual order in case the two
numbers which are compared have the same parity:

0246810 . 13579...

An order of type omega+omega+omega: just decide that multiple of 3 are
bigger than the multiple of two, themselves bigger than the remaining
numbers:

15711131417...  0246810... 3691215...

Again it should be easy to write a Fortran program capable of deciding
that order (that is to decide for any x and y if x  y with that
(unusual) order.

Exercise: could you find an order of type omega*omega? (Hint: use the
prime numbers).

You could use the Sieve of Eratosthenes (spelling?):

2*12*22*3... 3*13*33*5(all multiples of 3 that are not multiples
of 2)...
5*15*55*7(all multiples of 5 that are not multiples of 3 or 2)...

It sounds like the cute theorem says that you can keep dividing up the
natural numbers like this forever.

Those omega names are quite standard.

OK.  So we haven't left the finite behind yet.  It makes intuitive
sense to me that you can diagonalize till the cows come home, staying
within countability, and still not be done.  Otherwise infinity
wouldn't be infinite.

On the tricky question, it also makes intuitive sense that you can
sequence effectively on all computable growing functions.  This is
because the larger the growing function gets, the more uncovered space
(gaps) there are between the computable functions.  Any scheme for
generating growing functions will also leave behind every-growing
uncomputed gaps.  Very unmathematical of me to be so vague, but you've
already given us the answer, and I know you will fill in the gaps.  :)

I will. Unfortunately this week is full of duty charges. Meanwhile, I
would like to ask George and the others if they have a good
understanding of the present thread, that is on the fact that growing
functions has been well defined, that each sequence of such functions
are well defined, and each diagonalisation defines quite well a precise
programmable growing function (growing faster than the one in the
sequence it comes from).
Just a tiny effort, and I think we will have all we need to go into the
heart of the matter, and to understand why comp makes our universe
a godelized one in the Smullyan sense.

I meant that it makes intuitive sense that you *cannot* sequence
effectively on all computable growing functions.

You are right. But would that mean we cannot dovetail on all growing
computable functions? I let you ponder this not so easy question.

Bruno

PS About Parfit, I have already said some time ago in this list that I
appreciate very much its Reasons and Persons book, but, in the middle
of the book he makes the statement that we are token, where it
follows---[easily? not really: you need the movie graph or some strong
form of Occam]--- that comp makes us type, even abstract type. It just
happens that from a first person point of view we cannot take ourselves
as type because we just cannot distinguish between our many instances
generated by the Universal Dovetailer. A similar phenomenon occur
already with the quantum. But from the point of view of this thread,
this is an anticipation. The things which look the more like token,
with the comp hyp, are the numbers. This makes the second half part of
Parfit's book rather inconsistent with comp, but, still, his analysis
of personal identity remains largely genuine. (I don't like at all his
use of the name reductionism in that context, also, it's quite

http://iridia.ulb.ac.be/~marchal/

From what you've said about dovetailing before, you don't have to have
just a single sequence in order to dovetail.  You can jump among
multiple sequences.  I have yet to understand how you could dovetail on
something that is not effective.  That seems to be where 

### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


Tom Caylor wrote:

It sounds like the cute theorem says that you can keep dividing up the
natural numbers like this forever.

Oops.  I slipped in an actual infinity when I said forever.  Perhaps
I should have said indefinitely  ;)

Tom

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### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


Hi,

From what you've said about dovetailing before, you don't have to have

just a single sequence in order to dovetail.  You can jump among
multiple sequences.  I have yet to understand how you could dovetail on
something that is not effective.

I think dovetailing is possible because the dovetailer only complete sequences
at infinity. So when you construct the matrice on which you
will diagonalize, you are already diagonilizing it at the same time.
Example: when you have the first number of the first growing function, you
can also have the first number of the diagonalize function (by adding 1) and
the first number of the diagonalize*diagonalize function and ... ad recursum.
By dovetailing you execute in fact everything in parallel but all infinites
sequences are only completed at infinity.

Quentin Anciaux

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### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)



Quentin Anciaux wrote:
Hi,

From what you've said about dovetailing before, you don't have to have

just a single sequence in order to dovetail.  You can jump among
multiple sequences.  I have yet to understand how you could dovetail on
something that is not effective.

I think dovetailing is possible because the dovetailer only complete sequences
at infinity. So when you construct the matrice on which you
will diagonalize, you are already diagonilizing it at the same time.
Example: when you have the first number of the first growing function, you
can also have the first number of the diagonalize function (by adding 1) and
the first number of the diagonalize*diagonalize function and ... ad recursum.
By dovetailing you execute in fact everything in parallel but all infinites
sequences are only completed at infinity.

Quentin Anciaux

OK.  Thanks.  But so far we have done only effective diagonalization.
I'll follow along as Bruno goes step by step.  Also, it seems to me
even with non-effective diagonalization there will be another problem
to solve:  When we dovetail, how do we know we are getting sufficient
(which means indefinite) level of substitution in finite amount of
computation?  (Also, I am waiting for a good explanation of how Church
Thesis comes into this.)  Again, I'll wait for the step by step
argument.

Tom

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### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


Bruno Marchal wrote:

Meanwhile, I
would like to ask George and the others if they have a good
understanding of the present thread, that is on the fact that growing
functions has been well defined, that each sequence of such functions
are well defined, and each diagonalisation defines quite well a precise
programmable growing function (growing faster than the one in the
sequence it comes from).
Just a tiny effort, and I think we will have all we need to go into the
heart of the matter, and to understand why comp makes our universe
a godelized one in the Smullyan sense.

To speak only for myself,  I think I have a sufficient understanding of
the thread. Essentially you have shown that one cannot form a set of all
numbers/functions because given any set of numbers/functions it is
always possible, using diagonalization,  to generate new
numbers/functions: the Plenitude is too large to be a set. This leads to
a problem with the assumption of the existence of a Universal Dovetailer
whose purpose is to generate all functions. I hope this summary is accurate.

George

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### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


George Levy wrote:

Bruno Marchal wrote:

Meanwhile, I
would like to ask George and the others if they have a good
understanding of the present thread, that is on the fact that growing
functions has been well defined, that each sequence of such functions
are well defined, and each diagonalisation defines quite well a precise
programmable growing function (growing faster than the one in the
sequence it comes from).
Just a tiny effort, and I think we will have all we need to go into the
heart of the matter, and to understand why comp makes our universe
a godelized one in the Smullyan sense.

To speak only for myself,  I think I have a sufficient understanding of
the thread. Essentially you have shown that one cannot form a set of all
numbers/functions because given any set of numbers/functions it is
always possible, using diagonalization,  to generate new
numbers/functions: the Plenitude is too large to be a set. This leads to
a problem with the assumption of the existence of a Universal Dovetailer
whose purpose is to generate all functions. I hope this summary is
accurate.

George

The dovetailer is only supposed to generate all *computable* functions
though, correct? And the diagonalization of the (countable) set of all
computable functions would not itself be computable.

Jesse

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### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


Jesse Mazer writes:
The dovetailer is only supposed to generate all *computable* functions
though, correct? And the diagonalization of the (countable) set of all
computable functions would not itself be computable.

The dovetailer I know does not seem relevant to this discussion about
functions.  It generates programs, not functions.  For example, it
generates all 1 bit programs and runs each for one cycle; then generates
all 2 bit programs and runs each for 2 cycles; then generates all 3
bit programs and runs each for 3 cycles; and so on indefinitely.  (This
assumes that the 3 bit programs include all 2- and 1-bit programs, etc.)
In this way all programs get run with an arbitrary number of cycles.

These programs differ from functions in two ways.  First, programs may
never halt and hence may produce no fixed output, while functions must
have a well defined result.  And second, these programs take no inputs,
while functions should have at least one input variable.

What do you understand a dovetailer to be, in the context of computable
functions?

Hal Finney

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### Re: Smullyan Shmullyan, give me a real example


Rich,

are you familiar with the work of R.D. Laing? He was the illustrious
founder of the anti-psychiatry movement in the 60s. One never hears
of him these days. He had all the other thinkers on the hop for quite
a while. Your thoughts represent no interruption whatsoever.

Kim

On 29/05/2006, at 1:09 PM, Rich Winkel wrote:

At the risk of wasting more bandwidth than I alread have I'd like
to apologize for any discomfort I've caused on the list.   Sometimes
I feel like a jewish person arguing the reality of the holocaust
to doubters.  Such is the hidden record of psychiatry and the power
of its PR machine.  Please excuse the interruption.

Rich

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### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


Bruno Marchal wrote:
Le 26-mai-06, à 19:35, Tom Caylor a écrit :

Bruno,
You are starting to perturb me!  I guess that comes with the territory

You should not worry too much. I confess I am putting your mind in the
state of mathematicians before the Babbage Post Markov Turing Church
discovery. Everything here will be transparently clear.

But of course being perturbed doesn't
necessarily imply being correct.  I will summarize my perturbation
below.  But for now, specifically, you're bringing in transfinite
cardinals/ordinals.

Only transfinite ordinal which are all countable, and even nameable,
for example by name of growing computable functions as I am
illustrating.

Be sure you understand why G is a well defined computable growing
function, and why it grows faster than each initial Fi. If you know a
computer programming language, write the program!

This is where things get perverse and perhaps
inconsistent.  For instance, couldn't I argue that G is also infinite?

In which sense? All functions are infinite mathematical object.
Factorial is defined by its infiinite set of inputs outputs: {(0,1)
(1,1)(2,2) (3,6) (4,24) (5,120) ...}.

Take n = some fixed N1.  Then F1(N)  1, F2(N)  2, F3(N)  3, ...
and Fn(N)  n, for all n.  So each member of the whole sequence F1, F2,
F3 ... G is greater than the corresponding member of the sequence 1, 2,
3, ... aleph_0 (countable infinity).  Thus, G (=) countable infinity,
even for a fixed n=N1.

You are right but G is a function. Actually it just does what it has
been programmed to. I don't see any problem here.

OK.  I see that so far (above) there's no problem.  (See below for
where I still have concern(s).)  Here I was taking a fixed N, but G is
defined as the diagonal, so my comparison is not valid, and so my proof
that G is infinite for a fixed N is not valid.  I was taking G's
assignment of an ordinal of omega as being that it is in every way
larger than all Fi's, but in fact G is larger than all Fi's only when
comparing G(n) to Fn(n), not when comparing G(Nfixed) to Fi(N) for all
i's.

Oh Oh Oh Oh Oh  A new pattern emerge (the Ackerman Caylor one, at
a
higher level).

F_omega,
F_omega + omega
F_omega * omega
F_omega ^ omega
F_omega [4]  omega (omega tetrated to omega, actually this ordinal got
famous and is named Epsilon Zéro, will say some words on it later)

F_omega [5] omega
F_omega [6] omega
F_omega [7] omega
F_omega [8] omega
F_omega [9] omega
F_omega [10] omega
F_omega [11] omega

...

In this case they are all obtained by successive diagonalzations, but
nothing prevent us to diagonalise on it again to get

F_omega [omega] omega

OK, I think the following finite number is big enough:

F_omega [omega] omega (F_omega [omega] omega (9 [9] 9))

Next, we will meet a less constructivist fairy, and take some new kind
of big leap.

Be sure to be convinced that, despite the transfinite character of the
F_alpha sequence, we did really defined at all steps precise

It seems to me that you are on very shaky ground if you are citing
argument.

Please Tom, I did stay in the realm of the finitary. Even intutionist
can accept and prove correct the way I named what are just big finite
number. I have not until now transgressed the constructive field, I
have not begin to use Platonism! There is nothing controversial here,
even finitist mathematician can accept this. (Not ultra-finitist,
though, but those reject already 10^100)

I also think that if you could keep your arguments totally
in the finite arena it would less risky.

I have. You must (re)analyse the construction carefully and realize I
have not go out of the finite arena. Ordinals are just been used as a
way to put order on the successive effective diagonalizations. Those
are defined on perfectly well defined and generable sequence of well
defined functions. I have really just written a program (a little bit
sketchily, but you should be able to add the details once you should a
programming language).

OK.  I think you are just throwing me off with your notation.  Do you
have to use transfinite ordinals (omega) to do this?  Couldn't you just
stay with successively combining and diagonalizing, like below, without
using omegas?

G(n) = Fn(n)+1
Gi(n) = G(...G(n)), G taken i times
subscripts...
H1(n) = Gn(n)+1
H1i(n) = H1(...H1(n)), H1 taken i times
H2(n) = H1n(n)+1
H2i(n) = H2(...H2(n)), H2 taken i times

Then the subscripts count the number of diagonalizations you've done,
and every time you do the Ackermann thing, instead of adding an omega
do the Ackermann thing with the *number* of subscripts, 

### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


I meant that it makes intuitive sense that you *cannot* sequence
effectively on all computable growing functions.

Tom

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### Re: Smullyan Shmullyan, give me a real example


Thanks for that, Jesse. History-by-Hollywood has been my downfall
before...scriptwriter Akiva Goldsman should perhaps get six cuts of
the school cane for using such a high degree of creative licence with
the facts. A Best Movie vote seems to screw up one's scholarly
instincts to check the truth of the matter.  Goldsman has been
mercifully truer to the original with his job on The Da Vinci Code ;)

Nevertheless, do we see any interest in Nash's flitting in and out of
sanity and the corresponding strengthening and weakening of his
maths genius? As a reasonable (= non-schizo) machine he seemed
capable of a form of almost transcendental mathematical perception -
possibly almost as fabulous as Bruno's :) - during his extensive
schizophrenic interludes, a different form of perception seemed to
hold sway. Was he tripping between parallel versions of himself - or
what? The relevance to the discussion (for me) lies in his apparent
inabilty to choose his mental orientation. In both (opposing) mental
states he was being true to himself. He was surely a plural-self.
Maybe the imposition of a certain chemistry is the key to a
multiversal (ie non-linear) perception of self. Don't experiment with
this thought at home, though.

Kim

On 28/05/2006, at 11:18 AM, Jesse Mazer wrote:

Kim Jones wrote:

Well, in the case of schizoid mathematician John Nash, his
psychotic behaviour was also clearly linked to his maths ability.
After imbibing anti-psychotic medication, not only did his unreal
friends disappear, but his mathematical perception as well.

I don't think that's true, my understanding is that once he became
schizophrenic he no longer did any useful mathematical work, just
mystical
numerology. In discussing the movie, the wikipedia entry at
http://en.wikipedia.org/wiki/A_Beautiful_Mind says:

The movie also misrepresents the effect Nash's mental illness had
on his
work. The movie depicts Nash as already suffering from
schizophrenia when he
wrote his doctoral thesis. In reality, Nash's schizophrenia did not
appear
until years later and once it did his mathematical work ceased
until he was
able to bring it under control.

And the page at http://www.pnas.org/misc/classics5.shtml says that
he once
again started doing useful work after his recovery:

In 1970, Nash moved back to Princeton, where he took to shuffling
through
the halls of the mathematics building, occasionally scribbling
enigmatic
numerological messages on the walls. Students referred to him as the
Phantom of Fine Hall.

Gradually, however, Nash's mental condition began to improve.
Schizophrenia
rarely disappears completely, but by the 1990s Nash appeared to
remarkable recovery, and he had turned once again to mathematical
research.

The wikipedia article elaborates on what his recent work has been

The 1990s brought a return of his genius, and Nash has taken care
to manage
the symptoms of his mental illness. He is still hoping to score
substantial
scientific results. His recent work involves ventures in advanced game
theory including partial agency which show that, as in his early
career, he
prefers to select his own path and problems (though he continues to
work in
a communal setting to assist in managing his illness).

Jesse

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### RE: Smullyan Shmullyan, give me a real example


John M writes:

Stathis:
I call reasonable the items matching OUR (human) logic, even if we
call
it
a machine. There is no norm in the existence for 'reasonable', as
Cohen
and
Stewart showed in their chef d'oeuvre on Chaos in the imaginary
Zarathustrans.  We, with our 100 years ahead thinking and Bruno with
his
200 should be above such narrowminded limitations.
2.to your 'delusion': it is correctG.

Yes, but even with a normative definition of rational(if that is what
you mean) there are those who are either unable (due to illness, or
because they are infants, for example) or unwilling (due to expediency,
or laziness, or whatever) to be rational. Moreover, this is very common,
not just an exception to the rule.

)...The single best test is to treat someone with
antipsychotic medication and see if the delusion goes away.)
is this to implant new delusions and see how the poor fellow reacts?
at
al.) and so crazy is 'normal' and the norm may be crazy. Are the
psych-professionals exceptions?

Let me give you a real example. A patient claims that in the last month,
whenever he is away, or asleep, someone comes into his house and does
annoying things, like shifting personal items from one place to another,
putting holes in his socks, opening cupboards that were closed or
closing cupboards that were open, and so on. The culprit must know a lot
about electronic surveillance, because the cameras the poor victim has
installed do not show evidence of intrusion, and the neighbours might
also be involved because they all say they have seen nothing unusual,
even though it is happening almost every day. The patient stops going to
work or socializing, and drinks endless cups of coffee in order to stay
up as long as possible and catch the intruder, but it's no use: the
incidents become more frequent and more brazen. Finally, when the
patient starts talking about homicidal and suicidal impulses, his family
contact the local psychiatric crisis team, who make a diagnosis of
schizophreniform psychosis and persuade the patient to start taking
aripiprazole 15mg mane, which they supervise by visiting daily. The
patient only agrees to this because the alternative would be involuntary
hospital admission: he can explain, in great detail, why he is convinced
that the intruder is real, and to be fair, there is no way the
psychiatric team can prove him wrong, or be completely certain that he
is wrong.

Two weeks later, the alleged intrusions have almost completely stopped,
and our patient is thinking about going back to work again. He argues it
has nothing to do with the medication: maybe the culprit was scared off
by the psychiatric team, or maybe he just got sick of annoying him.

Two years later, the patient has gone through a cycle of stopping and
starting the medication half a dozen times. Within a few weeks of
stopping, the intrusions start up again, and within a few weeks of
recommencing treatment, they stop. He still isn't convinced that he has
ever been paranoid, but to stop his family and psychiatric services from
nagging him, he agrees to stay on the medication indefinitely, and he
remains well.

Given this story, are you still prepared to say that the patient's
reasoning that strangers are putting holes in his socks is just as valid
as that of the psychiatric team or his concerned family? It's worth
noting that his *deductive* reasoning remained intact throughout, so if
you could use this to explain logically where his *inductive* reasoning
went wrong, you might have saved the state the cost of an expensive
antipsychotic.

3. You wrote:
An unreasonable machine would look like a brain. The minds of living
organisms, such as they are, evolved ...
Because we know so little about the ways a brain works and assume too
much
based on our present ignorance to explain everything still unknown.
There
is
the terror of physicists forcing their primitive model on the world,
especially on domains where SOME features can be measured in
established
'phisics-invented' concepts by the so fa physics-invented
instruments
and
read in physics-invented units, although the conclusions come from
'non-physics-related' activities (mentality, ideation, feelings,
delusions, etc.,) all having parallel and physically measurable
phenomena
in the neurological sciences.
we use the 'brain' as a tool and have no idea how it works and for
what.

In your quoted fragment I feel an equating of brain and mind, which I
find
at least premature. I don't know what a mind may be. I know(?) it
must
be both atemporal and aspatial, while the material of the brain is
imagined
(physically) to be space and time related.

The mind is not the same as the brain, of course, but the two are
connected. We know that whenever certain complex physical processes
which we call brain activity occur, certain other mysterious 

### RE: Re: Smullyan Shmullyan, give me a real example



It's true that antipsychotics sometimes make people stupid; also fat, lazy, and any of about 10,000 other reported side-effects. Moreover, in most cases they don't bring about complete resolution of psychotic symptoms, and in some cases they seem to make little difference at all. Schizophrenia is a bad illness to get; but so is cancer, ischaemic heart disease, AIDS etc. etc., and the treatment for those is no more effective and no lesstoxic than the treatment of schizophrenia. Onthe other hand,about 1/3 of people who present with psychotic symptoms will have either a complete or a near-complete response to medication with minimal side-effects, and it would be tragic if they missed out due to anti-psychiatry prejudice or (more commonly) because they don't believe they have an illness.

There is a theory that the genes predisposing to schizophrenia have survived because they also give rise to original thinkers, conferring an advantage to society which outweighs the disadvantage of having floridly psychotic people around. According to this theory, it is only the *slightly* crazy ideas in those who don't develop the full-blown illness, or in Nash's case before they develop the full-blown illness, that are useful. The most likely outcome in an untreated floridly psychotic person who has to fend for himself is death.

Stathis Papaioannou

CC: everything-list@googlegroups.com From: [EMAIL PROTECTED] Subject: Re: Smullyan Shmullyan, give me a real example Date: Sun, 28 May 2006 10:52:32 +1000 To: [EMAIL PROTECTED]  Well,inthecaseofschizoidmathematicianJohnNash,his "psychotic"behaviourwasalsoclearlylinkedtohismathsability. Afterimbibinganti-psychoticmedication,notonlydidhis"unreal" friendsdisappear,buthismathematicalperceptionaswell.Thebind hefoundhimselfinwassurelythentobeatonceanunreasonable machine(underyoursandBruno'sdefinition)andareasonablemachine aswell-andtobebothsimultaneously!!!ForNash,thedelusional wasthedoorwaytoprovability.Hecouldnotseparatethetwo,except undertheinfluenceofheavychemistry.Canwedoanybetter?Should weeventry?  Kim   On27/05/2006,at10:25PM,StathisPapaioannouwrote:  Itisinterestingthatinpsychiatry,itisimpossibletogivea reliablemethodforrecognizingadelusion.Theusualdefinitionis that adelusionisafixed,falsebeliefwhichisnotinkeepingwiththe patient'sculturalbackground.Ifyouthinkaboutit,whyshould culturalbackgroundhaveanybearingonwhetheraperson's reasoningis faulty?Andevenincludingthiscriterion,itisoftendifficultto tell withoutlookingatassociatedfactorssuchaschangeinpersonality, mooddisturbance,etc.Thesinglebesttestistotreatsomeonewith antipsychoticmedicationandseeifthedelusiongoesaway.Express yourself instantly with MSN Messenger! MSN Messenger
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### Re: Re: Smullyan Shmullyan, give me a real example


According to Stathis Papaioannou:
On the other hand, about 1/3 of people who present with psychotic
symptoms will have either a complete or a near-complete response
to medication with minimal side-effects, and it would be tragic if
they missed out due to anti-psychiatry prejudice or (more commonly)
because they don't believe they have an illness.

Please don't bait me or the other unfortunates who have good reasons
for anti-psychiatry prejudice.  If psychiatry had any insight into its
own ignorance and destructiveness there'd be no reason for such prejudice.
Tweaking brain chemicals or running house current through brains
in pursuit of behavior modification and perception management is
like trying to program a computer with a soldering iron.  Selling
such quackery as medicine against nonconsenting children while in the
pay of their parents or other conflicted adults is an atrocity.
Shrinks have forgotten the principle of first no harm.

There is a theory that the genes predisposing to schizophrenia have
survived because they also give rise to original thinkers, conferring
floridly psychotic people around.

You're jumping between individual and meta-social fitness here. Where
is the feedback loop?

Rich

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### Re: Re: Smullyan Shmullyan, give me a real example


At the risk of wasting more bandwidth than I alread have I'd like
to apologize for any discomfort I've caused on the list.   Sometimes
I feel like a jewish person arguing the reality of the holocaust
to doubters.  Such is the hidden record of psychiatry and the power
of its PR machine.  Please excuse the interruption.

Rich

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### RE: Smullyan Shmullyan, give me a real example


Kim Jones writes:

Bruno,

what would an unreasonable machine be like? You seem to be implying
they exist, also that they can prove things about their possible
neighborhoods and or histories. (?)

Kim

An unreasonable machine would look like a brain. The minds of living
organisms, such as they are, evolved to promote survival and
reproduction, and apparently being rational is only a minor advantage
towards this end. I am sure that even logicians, at least when they are
off duty, pluck axioms out of the air according to whim or fashion, hold
contradictory beliefs simultaneously or sequentially, decide that the
correct course of action is x and then do ~x anyway, and so on.

It is interesting that in psychiatry, it is impossible to give a
reliable method for recognizing a delusion. The usual definition is that
a delusion is a fixed, false belief which is not in keeping with the
patient's cultural background. If you think about it, why should
cultural background have any bearing on whether a person's reasoning is
faulty? And even including this criterion, it is often difficult to tell
without looking at associated factors such as change in personality,
mood disturbance, etc. The single best test is to treat someone with
antipsychotic medication and see if the delusion goes away. This means
that in theory there might be two people with exactly the same belief,
justified in exactly the same way, but one is demonstrably psychotic
while the other is not! Crazy thinking is so common that, by itself, it
is generally not enough reason to diagnose someone as being crazy.

Stathis Papaioannou

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### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)



Le 26-mai-06, à 19:35, Tom Caylor a écrit :

Bruno Marchal wrote:
Hi,

OK, let us try to name the biggest natural (finite) number we can, and
let us do that transfinite ascension on the growing functions from N
to
N.

We have already build some well defined sequence of description (code)
of growing functions.

Let us choose the Hall Finney sequence to begin with (but the one by
Tom Caylor can be use instead).

F1 F2 F3 F4 F5 ...

With F1(n) = factorial(n), F2(n) = factorial(factorial n), etc.

Note this: Hal gave us a trick for getting from a growing function f,
a
new function growing faster, actually the iteration of the function.
That is, Hal gave us a notion of successor for the growing function.
Now the diagonalization of the sequence F1 F2 F3 F4 ..., which is
given
by the new growing function defined by

G(n) = Fn(n) + 1

gives us a growing function which grows faster than any Fi from Hal's
initial sequence. Precisely, G will grow faster than any Fi on *almost
all* number (it could be that some Fi will grow faster than G on some
initial part of N, but for some finite value (which one?) G will keep
growing faster. Technically we must remember to apply our growing
function on sufficiently big input' if we want to benefit of the
growing phenomenon. We will make a rough evaluation on that input
later, but let us not being distract by technical point like that.
The diagonalization gives an effective way to take the limit of the
sequence F1, F2, F3, ...

G grows faster than any Fi. Mathematician will say that the order type
of g, in our our new sequence F1 F2 F3 ... G,  is omega (the greek
letter).

Bruno,
You are starting to perturb me!  I guess that comes with the territory

You should not worry too much. I confess I am putting your mind in the
state of mathematicians before the Babbage Post Markov Turing Church
discovery. Everything here will be transparently clear.

But of course being perturbed doesn't
necessarily imply being correct.  I will summarize my perturbation
below.  But for now, specifically, you're bringing in transfinite
cardinals/ordinals.

Only transfinite ordinal which are all countable, and even nameable,
for example by name of growing computable functions as I am
illustrating.

Be sure you understand why G is a well defined computable growing
function, and why it grows faster than each initial Fi. If you know a
computer programming language, write the program!

This is where things get perverse and perhaps
inconsistent.  For instance, couldn't I argue that G is also infinite?

In which sense? All functions are infinite mathematical object.
Factorial is defined by its infiinite set of inputs outputs: {(0,1)
(1,1)(2,2) (3,6) (4,24) (5,120) ...}.

Take n = some fixed N1.  Then F1(N)  1, F2(N)  2, F3(N)  3, ...
and Fn(N)  n, for all n.  So each member of the whole sequence F1, F2,
F3 ... G is greater than the corresponding member of the sequence 1, 2,
3, ... aleph_0 (countable infinity).  Thus, G (=) countable infinity,
even for a fixed n=N1.

You are right but G is a function. Actually it just does what it has
been programmed to. I don't see any problem here.

But G is just a well defined computable growing function and we can
use
Hall Finney successor again to get the still faster function, namely
G(G(n)).

The order type of G(G(n)) is, well, the successor of omega: omega+1

And, as Hall initially, we can build the new sequance of growing
functions (all of which grows more than the preceding sequence):

G(n) G(G(n)) G(G(G(n))) G(G(G(G(n etc.

which are of order type omega, omega+1, omega+2, omega+3, omega+4,
etc.

Now we have obtained a new well defined infinite sequence of growing
function, and, writing it as:

G1, G2, G3, G4, G5, G6, ...  or better, as

F_omega, F_omega+1, F_omega+2, F_omega+3

just showing such a sequence can be generated so that we can again
diagonalise it, getting

H(n) = Gn(n) + 1, or better

H(n) = F_omega+n (n) + 1

Getting a function of order type omega+omega: we can write H =
F_omega+omega

And of course, we can apply Hall's successor again, getting
F_omega+omega+1
which is just H(H(n), and so we get a new sequence:

F_omega+omega+1, F_omega+omega+2, F_omega+omega+3, ...

Which can be diagonalise again, so we get

F_omega+omega+omega,

and then by Hal again, and again ...:

F_omega+omega+omega+1, F_omega+omega+omega+2, F_omega+omega+omega+3

...

Oh Oh! a new pattern emerges, a new type of sequence of well defined
growing functions appears:

F_omega, F_omega+omega, F_omega+omega+omega,
F_omega+omega+omega+omega,

And we can generated it computationnaly, so we can diagonalise again
to
get:

F_omega * times omega,

and of course we can apply Hal's successor (or caylor one of course)
again, and again

Oh Oh Oh Oh Oh  A new pattern emerge (the Ackerman Caylor one, at
a
higher 

### Re: Smullyan Shmullyan, give me a real example


Stathis:
I call reasonable the items matching OUR (human) logic, even if we call it
a machine. There is no norm in the existence for 'reasonable', as Cohen and
Stewart showed in their chef d'oeuvre on Chaos in the imaginary
Zarathustrans.  We, with our 100 years ahead thinking and Bruno with his
200 should be above such narrowminded limitations.
2.to your 'delusion': it is correctG.
)...The single best test is to treat someone with
antipsychotic medication and see if the delusion goes away.)
is this to implant new delusions and see how the poor fellow reacts?
al.) and so crazy is 'normal' and the norm may be crazy. Are the
psych-professionals exceptions?
3. You wrote:
An unreasonable machine would look like a brain. The minds of living
organisms, such as they are, evolved ...
Because we know so little about the ways a brain works and assume too much
based on our present ignorance to explain everything still unknown. There is
the terror of physicists forcing their primitive model on the world,
especially on domains where SOME features can be measured in established
'phisics-invented' concepts by the so fa physics-invented instruments and
read in physics-invented units, although the conclusions come from
'non-physics-related' activities (mentality, ideation, feelings,
delusions, etc.,) all having parallel and physically measurable phenomena
in the neurological sciences.
we use the 'brain' as a tool and have no idea how it works and for what.

In your quoted fragment I feel an equating of brain and mind, which I find
at least premature. I don't know what a mind may be. I know(?) it must
be both atemporal and aspatial, while the material of the brain is imagined
(physically) to be space and time related.

John M
- Original Message -
From: Stathis Papaioannou [EMAIL PROTECTED]
Cc: [EMAIL PROTECTED]
Sent: Saturday, May 27, 2006 8:25 AM
Subject: RE: Smullyan Shmullyan, give me a real example

Kim Jones writes:

Bruno,

what would an unreasonable machine be like? You seem to be implying
they exist, also that they can prove things about their possible
neighborhoods and or histories. (?)

Kim

An unreasonable machine would look like a brain. The minds of living
organisms, such as they are, evolved to promote survival and
reproduction, and apparently being rational is only a minor advantage
towards this end. I am sure that even logicians, at least when they are
off duty, pluck axioms out of the air according to whim or fashion, hold
contradictory beliefs simultaneously or sequentially, decide that the
correct course of action is x and then do ~x anyway, and so on.

It is interesting that in psychiatry, it is impossible to give a
reliable method for recognizing a delusion. The usual definition is that
a delusion is a fixed, false belief which is not in keeping with the
patient's cultural background. If you think about it, why should
cultural background have any bearing on whether a person's reasoning is
faulty? And even including this criterion, it is often difficult to tell
without looking at associated factors such as change in personality,
mood disturbance, etc. The single best test is to treat someone with
antipsychotic medication and see if the delusion goes away. This means
that in theory there might be two people with exactly the same belief,
justified in exactly the same way, but one is demonstrably psychotic
while the other is not! Crazy thinking is so common that, by itself, it
is generally not enough reason to diagnose someone as being crazy.

Stathis Papaioannou

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### Re: Smullyan Shmullyan, give me a real example


Well, in the case of schizoid mathematician John Nash, his
psychotic behaviour was also clearly linked to his maths ability.
After imbibing anti-psychotic medication, not only did his unreal
friends disappear, but his mathematical perception as well. The bind
he found himself in was surely then to be at once an unreasonable
machine (under yours and Bruno's definition) and a reasonable machine
as well - and to be both simultaneously!!! For Nash, the delusional
was the doorway to provability. He could not separate the two, except
under the influence of heavy chemistry. Can we do any better? Should
we even try?

Kim

On 27/05/2006, at 10:25 PM, Stathis Papaioannou wrote:

It is interesting that in psychiatry, it is impossible to give a
reliable method for recognizing a delusion. The usual definition is
that
a delusion is a fixed, false belief which is not in keeping with the
patient's cultural background. If you think about it, why should
cultural background have any bearing on whether a person's
reasoning is
faulty? And even including this criterion, it is often difficult to
tell
without looking at associated factors such as change in personality,
mood disturbance, etc. The single best test is to treat someone with
antipsychotic medication and see if the delusion goes away.

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### Re: Smullyan Shmullyan, give me a real example


Kim Jones wrote:

Well, in the case of schizoid mathematician John Nash, his
psychotic behaviour was also clearly linked to his maths ability.
After imbibing anti-psychotic medication, not only did his unreal
friends disappear, but his mathematical perception as well.

I don't think that's true, my understanding is that once he became
schizophrenic he no longer did any useful mathematical work, just mystical
numerology. In discussing the movie, the wikipedia entry at
http://en.wikipedia.org/wiki/A_Beautiful_Mind says:

The movie also misrepresents the effect Nash's mental illness had on his
work. The movie depicts Nash as already suffering from schizophrenia when he
wrote his doctoral thesis. In reality, Nash's schizophrenia did not appear
until years later and once it did his mathematical work ceased until he was
able to bring it under control.

And the page at http://www.pnas.org/misc/classics5.shtml says that he once
again started doing useful work after his recovery:

In 1970, Nash moved back to Princeton, where he took to shuffling through
the halls of the mathematics building, occasionally scribbling enigmatic
numerological messages on the walls. Students referred to him as the
Phantom of Fine Hall.

Gradually, however, Nash's mental condition began to improve. Schizophrenia
rarely disappears completely, but by the 1990s Nash appeared to have made a
remarkable recovery, and he had turned once again to mathematical research.

The wikipedia article elaborates on what his recent work has been about:

The 1990s brought a return of his genius, and Nash has taken care to manage
the symptoms of his mental illness. He is still hoping to score substantial
scientific results. His recent work involves ventures in advanced game
theory including partial agency which show that, as in his early career, he
prefers to select his own path and problems (though he continues to work in
a communal setting to assist in managing his illness).

Jesse

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### Re: Smullyan Shmullyan, give me a real example



Le 26-mai-06, à 02:50, James N Rose a écrit :

Bruno,

You struck a personal nerve in me with your following remarks:

Bruno Marchal wrote:

They are degrees. The worst unreasonableness of a (platonist or
classical or even intuitionist) machine is when she believes some
plain
falsity (like p  ~p, or 0 = 1). The false implies all propositions,
so
that such machine believes everything, including everything about
their
maximal consistent extensions or histories (which does not exist).
Those machines are just inconsistent.

particularly ,

some plain falsity (like p  ~p, or 0 = 1).

Rather than treat these as 'blatantly false' I have been
exploring the notion for several years .. 'what conditions,
situations, criteria or states would allow such statements
to be 'true', and what would it mean in how we define and
manipulate and operate the rest of mathematics?'.

I have discovered that an unprecedentedly un-appreciated
realm of mathematical relations has existed right before
our minds.  The lack, having kept us trying to cope with
'anomalies' and math issues without the full toolkit of
mathematical instruments.

An example at the core of it is a most simplistic
definition/equation.

1^1 = 1^0

[one to the exponent one  equals  one to the exponent zero]

To all mathematicians, this is a toss-out absurdity, with
no 'real meaning'.  n^0 is a convenience tool at best ;

n^0 = 1, because 1= (n^m)/(n^m) = n^(m-m) = n^0.
Or better n^0 = the number of functions from the empty set (cardinal 0)
to the
set with cardinal n. This justifies also 0^0 = 1 (there is one (empty)
function from
the empty set to the empty set).

along
with  'n/0 is 'undefined''.   We note the consistent/valid
notation, but walk away from any active utility or application.

My thesis is that doing so was a missed opportunity.

To be hyper-consistent, the equation set-up

1^1 = 1^0

indicates that there -must- be some valid states/conditions
(not just 'interpretation') when 0 and 1 are 'equal' in some
real meaning/use of the word equal.

Why? It is usual that a function (like y = 1^x) can have the same value
for different argument.
From (-5)^2 = 5^2 you will not infer that 5 = (-5), right?
From sinus(x) = sinus(pi - x) you will not deduce that x = pi - x,
right?

If they can be substituted
in the above equation, without changing a resultant of
calculations (they are embedded in), then they must somewhere
somehow in fact be identical in some way or condition.

You talk like if all functions are bijections (one to one function).

The entire ediface of physics is hamstrung because of this,
because mathematical definitions and language compounded
the error by applying - actually DIS-applying - a related
concept .. the notion of 'extent' .. also known as 'dimension'.

Physics and mathematics transform and wholly open up when
we throw away the old concept of 'dimensionless' and instead
reformulate -everything- as 'dimensional'.  Including zero;
including numbers unassociated with variables.

As musch as you are brilliant and mathematically inventive,
your statement  some plain falsity (like p  ~p, or 0 = 1)
shows you haven't quite awoken to everything yet.  I hope
I'm in the process of stirring you from your slumber.

I am using the name 0, 1, ... for the usual numbers. 1 is different
from 0 for the same reason that 1 cup of coffee is different from 0 cup
of coffee, or that 1 joke is different from 0 joke ...

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)


Bruno Marchal wrote:
Hi,

OK, let us try to name the biggest natural (finite) number we can, and
let us do that transfinite ascension on the growing functions from N to
N.

We have already build some well defined sequence of description (code)
of growing functions.

Let us choose the Hall Finney sequence to begin with (but the one by
Tom Caylor can be use instead).

F1 F2 F3 F4 F5 ...

With F1(n) = factorial(n), F2(n) = factorial(factorial n), etc.

Note this: Hal gave us a trick for getting from a growing function f, a
new function growing faster, actually the iteration of the function.
That is, Hal gave us a notion of successor for the growing function.
Now the diagonalization of the sequence F1 F2 F3 F4 ..., which is given
by the new growing function defined by

G(n) = Fn(n) + 1

gives us a growing function which grows faster than any Fi from Hal's
initial sequence. Precisely, G will grow faster than any Fi on *almost
all* number (it could be that some Fi will grow faster than G on some
initial part of N, but for some finite value (which one?) G will keep
growing faster. Technically we must remember to apply our growing
function on sufficiently big input' if we want to benefit of the
growing phenomenon. We will make a rough evaluation on that input
later, but let us not being distract by technical point like that.
The diagonalization gives an effective way to take the limit of the
sequence F1, F2, F3, ...

G grows faster than any Fi. Mathematician will say that the order type
of g, in our our new sequence F1 F2 F3 ... G,  is omega (the greek
letter).

Bruno,
You are starting to perturb me!  I guess that comes with the territory
where you're leading us.  But of course being perturbed doesn't
necessarily imply being correct.  I will summarize my perturbation
below.  But for now, specifically, you're bringing in transfinite
cardinals/ordinals.  This is where things get perverse and perhaps
inconsistent.  For instance, couldn't I argue that G is also infinite?
Take n = some fixed N1.  Then F1(N)  1, F2(N)  2, F3(N)  3, ...
and Fn(N)  n, for all n.  So each member of the whole sequence F1, F2,
F3 ... G is greater than the corresponding member of the sequence 1, 2,
3, ... aleph_0 (countable infinity).  Thus, G (=) countable infinity,
even for a fixed n=N1.

But G is just a well defined computable growing function and we can use
Hall Finney successor again to get the still faster function, namely
G(G(n)).

The order type of G(G(n)) is, well, the successor of omega: omega+1

And, as Hall initially, we can build the new sequance of growing
functions (all of which grows more than the preceding sequence):

G(n) G(G(n)) G(G(G(n))) G(G(G(G(n etc.

which are of order type omega, omega+1, omega+2, omega+3, omega+4, etc.

Now we have obtained a new well defined infinite sequence of growing
function, and, writing it as:

G1, G2, G3, G4, G5, G6, ...  or better, as

F_omega, F_omega+1, F_omega+2, F_omega+3

just showing such a sequence can be generated so that we can again
diagonalise it, getting

H(n) = Gn(n) + 1, or better

H(n) = F_omega+n (n) + 1

Getting a function of order type omega+omega: we can write H =
F_omega+omega

And of course, we can apply Hall's successor again, getting
F_omega+omega+1
which is just H(H(n), and so we get a new sequence:

F_omega+omega+1, F_omega+omega+2, F_omega+omega+3, ...

Which can be diagonalise again, so we get

F_omega+omega+omega,

and then by Hal again, and again ...:

F_omega+omega+omega+1, F_omega+omega+omega+2, F_omega+omega+omega+3

...

Oh Oh! a new pattern emerges, a new type of sequence of well defined
growing functions appears:

F_omega, F_omega+omega, F_omega+omega+omega, F_omega+omega+omega+omega,

And we can generated it computationnaly, so we can diagonalise again to
get:

F_omega * times omega,

and of course we can apply Hal's successor (or caylor one of course)
again, and again

Oh Oh Oh Oh Oh  A new pattern emerge (the Ackerman Caylor one, at a
higher level).

F_omega,
F_omega + omega
F_omega * omega
F_omega ^ omega
F_omega [4]  omega (omega tetrated to omega, actually this ordinal got
famous and is named Epsilon Zéro, will say some words on it later)

F_omega [5] omega
F_omega [6] omega
F_omega [7] omega
F_omega [8] omega
F_omega [9] omega
F_omega [10] omega
F_omega [11] omega

...

In this case they are all obtained by successive diagonalzations, but
nothing prevent us to diagonalise on it again to get

F_omega [omega] omega

OK, I think the following finite number is big enough:

F_omega [omega] omega (F_omega [omega] omega (9 [9] 9))

Next, we will meet a less constructivist fairy, and take some new kind
of big leap.

Be sure to be convinced that, despite the transfinite character of the
F_alpha sequence, we did really defined at all steps precise

It seems to me that you are on very 

### Re: Smullyan Shmullyan, give me a real example


Bruno,

what would an unreasonable machine be like? You seem to be implying
they exist, also that they can prove things about their possible
neighborhoods and or histories. (?)

Kim

On 23/05/2006, at 8:25 PM, Bruno Marchal wrote:

Is it not utterly obvious that, IF we are (hopefully
reasonable) machine, THEN we will learn something genuine by studying
possible neighborhoods and or histories?

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### Re: Smullyan Shmullyan, give me a real example



Le 25-mai-06, à 09:04, Kim Jones a écrit :

what would an unreasonable machine be like? You seem to be implying
they exist, also that they can prove things about their possible
neighborhoods and or histories. (?)

They are degrees. The worst unreasonableness of a (platonist or
classical or even intuitionist) machine is when she believes some plain
falsity (like p  ~p, or 0 = 1). The false implies all propositions, so
that such machine believes everything, including everything about their
maximal consistent extensions or histories (which does not exist).
Those machines are just inconsistent.

Then you have machines which, although they are consistent, are not
self-referentially correct. They are unsound, and does also believe
some falsity, but here the falsity is irrefutable. Like consistent
machine asserting they are inconsistent, or, curiously enough (it is a
consequence of the second incompleteness theorem), consistent machine
asserting they are consistent. Although it is true that they are
consistent they cannot assert it without becoming either inconsistent,
or, if they assert it in some special cautious way, they become
different machine (and in that case they remain consistent and also get
some new provability power).

But again here we anticipate. I hope I will make clear that with Church
thesis the notion of computability will appear as absolute and
universal, and then we will see that the notion of provability is
relative and never universal, although some universal pattern can
appear there too (like the logic G and G*, etc.).

I guess we will come back on this, but we have to do a transfinite
ascension before :)

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example



Le 24-mai-06, à 18:30, Tom Caylor a écrit :

Exercises:

0) Could you evaluate roughly the number of digit of 4 [4] 4 ? What
about the number of digit of fact(fact(fact(fact 4

1) is the diagonal g function a growing function? Could g belong to
the
initial sequence, does g grows more quickly than any function in the
initial sequence, and in what sense precisely.

2) Could you find a function, and even a new sequence of functions
more
and more growing, and growing more than the function g?

3) Do you see why it is said that g is build by diagonalization? Where
is the diagonal?

4) Is there a universal sequence of growing functions, i. e.
containing
all computable growing functions?

Must already go. Sorry for this quick piece. Solution tomorrow. Hope
before missing the real start ... Any comments , critics or
suggestions
are welcome ...

Bruno

http://iridia.ulb.ac.be/~marchal/

I don't have time right now for detailed computations, but I'll give a

g is the same as my f(n,n,n)+1, and I already commented that f(n,m,n)
is a growing function, since f(N,m,n) is growing for fixed N.  So
clearly g is growing.  As I said about f(n,m,n), the degree or -ation
of g grows as n (or x) grows.  I recognize that adding 1 to make g is
the classical diagonalization move.  It makes g different from any Fi
in the sequence Fi, i=1,2,3,...  And in fact, since we add 1, rather
than subtract 1, g is larger than any Fi.

I'm having a problem with accepting g, or even my original f(n,n,n), as
a function in the same sense as with a fixed degree or -ation of
operation.  This is because the definition of the function changes
depending on the value taken in the domain of the function.  Is this
valid?

It is. Actually the definition of your f does not change, given that
you are using a parameter to capture that change.
It is valid because you did build a well defined computable function.
Actually Ackermann invented his function for showing the existence of a
computable function (from N to N) which does not belong to the already
very large (but not universal) set of so called primitive recursive
function.

However, if we just ignore this problem, throwing caution to the wind,
then the next logical iteration of diagonalization is to do the
-ation thing on g and then diagonalize.  Let Gi(x) = g(x) [i] g(x),
then let h(x) = Gx(x) + 1.

Good idea. And there is no reason to stop there in case the fairy give
you some more paper. See my next post.

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Ascension (was Re: Smullyan Shmullyan, give me a real example)


Hi,

OK, let us try to name the biggest natural (finite) number we can, and
let us do that transfinite ascension on the growing functions from N to
N.

We have already build some well defined sequence of description (code)
of growing functions.

Let us choose the Hall Finney sequence to begin with (but the one by
Tom Caylor can be use instead).

F1 F2 F3 F4 F5 ...

With F1(n) = factorial(n), F2(n) = factorial(factorial n), etc.

Note this: Hal gave us a trick for getting from a growing function f, a
new function growing faster, actually the iteration of the function.
That is, Hal gave us a notion of successor for the growing function.
Now the diagonalization of the sequence F1 F2 F3 F4 ..., which is given
by the new growing function defined by

G(n) = Fn(n) + 1

gives us a growing function which grows faster than any Fi from Hal's
initial sequence. Precisely, G will grow faster than any Fi on *almost
all* number (it could be that some Fi will grow faster than G on some
initial part of N, but for some finite value (which one?) G will keep
growing faster. Technically we must remember to apply our growing
function on sufficiently big input' if we want to benefit of the
growing phenomenon. We will make a rough evaluation on that input
later, but let us not being distract by technical point like that.
The diagonalization gives an effective way to take the limit of the
sequence F1, F2, F3, ...

G grows faster than any Fi. Mathematician will say that the order type
of g, in our our new sequence F1 F2 F3 ... G,  is omega (the greek
letter).

But G is just a well defined computable growing function and we can use
Hall Finney successor again to get the still faster function, namely
G(G(n)).

The order type of G(G(n)) is, well, the successor of omega: omega+1

And, as Hall initially, we can build the new sequance of growing
functions (all of which grows more than the preceding sequence):

G(n) G(G(n)) G(G(G(n))) G(G(G(G(n etc.

which are of order type omega, omega+1, omega+2, omega+3, omega+4, etc.

Now we have obtained a new well defined infinite sequence of growing
function, and, writing it as:

G1, G2, G3, G4, G5, G6, ...  or better, as

F_omega, F_omega+1, F_omega+2, F_omega+3

just showing such a sequence can be generated so that we can again
diagonalise it, getting

H(n) = Gn(n) + 1, or better

H(n) = F_omega+n (n) + 1

Getting a function of order type omega+omega: we can write H =
F_omega+omega

And of course, we can apply Hall's successor again, getting
F_omega+omega+1
which is just H(H(n), and so we get a new sequence:

F_omega+omega+1, F_omega+omega+2, F_omega+omega+3, ...

Which can be diagonalise again, so we get

F_omega+omega+omega,

and then by Hal again, and again ...:

F_omega+omega+omega+1, F_omega+omega+omega+2, F_omega+omega+omega+3

...

Oh Oh! a new pattern emerges, a new type of sequence of well defined
growing functions appears:

F_omega, F_omega+omega, F_omega+omega+omega, F_omega+omega+omega+omega,

And we can generated it computationnaly, so we can diagonalise again to
get:

F_omega * times omega,

and of course we can apply Hal's successor (or caylor one of course)
again, and again

Oh Oh Oh Oh Oh  A new pattern emerge (the Ackerman Caylor one, at a
higher level).

F_omega,
F_omega + omega
F_omega * omega
F_omega ^ omega
F_omega [4]  omega (omega tetrated to omega, actually this ordinal got
famous and is named Epsilon Zéro, will say some words on it later)

F_omega [5] omega
F_omega [6] omega
F_omega [7] omega
F_omega [8] omega
F_omega [9] omega
F_omega [10] omega
F_omega [11] omega

...

In this case they are all obtained by successive diagonalzations, but
nothing prevent us to diagonalise on it again to get

F_omega [omega] omega

OK, I think the following finite number is big enough:

F_omega [omega] omega (F_omega [omega] omega (9 [9] 9))

Next, we will meet a less constructivist fairy, and take some new kind
of big leap.

Be sure to be convinced that, despite the transfinite character of the
F_alpha sequence, we did really defined at all steps precise

Tricky Problem: is there a sequence in which all growing computable
functions belong? Is it possible to dovetail on all computable growing
functions, ...

I let you think,

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example


Bruno,

You struck a personal nerve in me with your following remarks:

Bruno Marchal wrote:

They are degrees. The worst unreasonableness of a (platonist or
classical or even intuitionist) machine is when she believes some plain
falsity (like p  ~p, or 0 = 1). The false implies all propositions, so
that such machine believes everything, including everything about their
maximal consistent extensions or histories (which does not exist).
Those machines are just inconsistent.

particularly ,

some plain falsity (like p  ~p, or 0 = 1).

Rather than treat these as 'blatantly false' I have been
exploring the notion for several years .. 'what conditions,
situations, criteria or states would allow such statements
to be 'true', and what would it mean in how we define and
manipulate and operate the rest of mathematics?'.

I have discovered that an unprecedentedly un-appreciated
realm of mathematical relations has existed right before
our minds.  The lack, having kept us trying to cope with
'anomalies' and math issues without the full toolkit of
mathematical instruments.

An example at the core of it is a most simplistic
definition/equation.

1^1 = 1^0

[one to the exponent one  equals  one to the exponent zero]

To all mathematicians, this is a toss-out absurdity, with
no 'real meaning'.  n^0 is a convenience tool at best ; along
with  'n/0 is 'undefined''.   We note the consistent/valid
notation, but walk away from any active utility or application.

My thesis is that doing so was a missed opportunity.

To be hyper-consistent, the equation set-up

1^1 = 1^0

indicates that there -must- be some valid states/conditions
(not just 'interpretation') when 0 and 1 are 'equal' in some
real meaning/use of the word equal.  If they can be substituted
in the above equation, without changing a resultant of
calculations (they are embedded in), then they must somewhere
somehow in fact be identical in some way or condition.

The entire ediface of physics is hamstrung because of this,
because mathematical definitions and language compounded
the error by applying - actually DIS-applying - a related
concept .. the notion of 'extent' .. also known as 'dimension'.

Physics and mathematics transform and wholly open up when
we throw away the old concept of 'dimensionless' and instead
reformulate -everything- as 'dimensional'.  Including zero;
including numbers unassociated with variables.

As musch as you are brilliant and mathematically inventive,
your statement  some plain falsity (like p  ~p, or 0 = 1)
shows you haven't quite awoken to everything yet.  I hope
I'm in the process of stirring you from your slumber.

Jamie Rose

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### Re: Smullyan Shmullyan, give me a real example



Hi Russell,

You wrote (24 may):

On Tue, May 23, 2006 at 12:25:35PM +0200, Bruno Marchal wrote:

In a sense, you are obviously right.  That is why I said some
knowledge of comp science or even just in math will make the existence
of the UD, and of the Universal Machine astonishing. Precisely it is
the knowledge of diagonalization. Godel will miss the universal
machine
and Church thesis, and will describe those things as a sort of
miracle.
More later.  I will comment again with much more detail the rest of
your post much later. If I comment it here now I will introduce
confusion. It is preferable people get much more familiarity with the
effective and not effective daigonalisations procedures before, I
think.

I guess by this you mean that whilst it is impossible enumerate all
descriptions (the books in the infinite version of the Library of
Babel such as I take as my starting point), nor all true mathematical
facts, or even all programs (not sure on this one, obviously one can
enumerate all halting programs), it is however possible to execute all
possible programs. Yes, put that way, I suppose it is astonishing.

You put your finger on the difficulty. If we can enumerate all halting
programs then we can diagonalize  it and extract an halting program not
belonging to the list.

Actually when you say in your preceding post (21 May):

That one can dovetail on all possible programs must be pretty obvious
once one realises that these can be enumerated.

You are pointing on the main difficulty. Once we can enumerate a list
of functions from N to N, then we can diagonalize that list, and by
this we can show the list being not complete.

Now, it is not so hard for a computer programmer to single out a
solution to this difficulty, but without a good understanding of
diagonalization, it is easy to miss what is going on, and to miss in
this way the tremendous impact of Church thesis. I will show that
Church thesis will literally rehabilitate Pythagorus doctrine: all is
number, despite irrational or transcendent numbers.

Let me proceed further with the others because we are far ahead in the

Best regards,

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example


Hi George, Tom, Hal, and others,

OK. I hope it is clear for everybody that, exactly like we have a
natural infinite sequence of positive integer or natural numbers:

0,  1,  2,  3,  4,  etc.

We have a natural sequence of growing functions, (also called
operations):

MULTIPLICATION
EXPONENTIATION
TETRATION
PENTATION
HEXATION
HEPTATION
OCTATION
ENNEATION
DECATION
11-ATION
12-ATION
TRISKAIDEKATION
14-ATION
15-ATION
16-ATION
17-ATION
...

(I remember the greek name of 13 thanks to the disease
triskadekaphobia : the fear of the number 13 :)

We can use the notation [n] for any n-ation, so that for example:

4 [1] 3 = 7,

4 [2] 3 = 12,

4 [3] 3 = 64,

4 [4] 3 =
134078079299425970995740249982058461274793658205923933777235614437217640
300735469768018742981669034276900318581864860508537538828119465699464336
49006084096,

4 [4] 4 = 4 ^ the preceding number [out-of-range of most computer
etc.

Let us write Fi(x) = x [i] x ; Indeed it will be more easy to
illustrate diagonalization on one variable function:

Thus F1(x) = x + x;  F2(x) = x * x, F3(x) = x ^ x, F4(x) = x [4] x,
F5(x) = x [5] x, F6(x) = x [6] x, etc.

This gives us an infinite list of one variable growing functions F0 F1,
F2, F3, F4, F5, F6, F7, ...

Please note that I could have taken Hal Finney list,  H0 H1 H2 H3 H4 H5
H6 H7 H8 H9 ...where H0(x) = factorial(x),
H1(x) =  factorial(factorial 2), H2(x) = factorial(factorial (factorial
(x))), ...

Mmmh... I am realizing it will even be easier to diagonalize
transfinitely with Hal Finney's functions than with the traditional
one, because with Hal Finney's one we will not been obliged of doing
some back and forth between one and two variable functions.

Anyway,  let

F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 ...

be your favorite sequence of one-variable more and more growing
function. (I recall all function here are function defined on N and
with value in N; where N = the set of natural numbers : 0, 1, 2, 3, ...

Here is a growing function, build from that class from diagonalization:

g(x) = Fx(x) + 1   (in english: to compute g(x), search the xth
function in your sequence, and apply it to x and then add 1.

For example  g(3) = F3(3) + 1, g(245) = F245(245) + 1, etc.

Exercises:

0) Could you evaluate roughly the number of digit of 4 [4] 4 ? What
about the number of digit of fact(fact(fact(fact 4

1) is the diagonal g function a growing function? Could g belong to the
initial sequence, does g grows more quickly than any function in the
initial sequence, and in what sense precisely.

2) Could you find a function, and even a new sequence of functions more
and more growing, and growing more than the function g?

3) Do you see why it is said that g is build by diagonalization? Where
is the diagonal?

4) Is there a universal sequence of growing functions, i. e. containing
all computable growing functions?

Must already go. Sorry for this quick piece. Solution tomorrow. Hope
before missing the real start ... Any comments , critics or suggestions
are welcome ...

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example



Bruno Marchal wrote:
Hi George, Tom, Hal, and others,

OK. I hope it is clear for everybody that, exactly like we have a
natural infinite sequence of positive integer or natural numbers:

0,  1,  2,  3,  4,  etc.

We have a natural sequence of growing functions, (also called
operations):

MULTIPLICATION
EXPONENTIATION
TETRATION
PENTATION
HEXATION
HEPTATION
OCTATION
ENNEATION
DECATION
11-ATION
12-ATION
TRISKAIDEKATION
14-ATION
15-ATION
16-ATION
17-ATION
...

(I remember the greek name of 13 thanks to the disease
triskadekaphobia : the fear of the number 13 :)

We can use the notation [n] for any n-ation, so that for example:

4 [1] 3 = 7,

4 [2] 3 = 12,

4 [3] 3 = 64,

4 [4] 3 =
134078079299425970995740249982058461274793658205923933777235614437217640
300735469768018742981669034276900318581864860508537538828119465699464336
49006084096,

4 [4] 4 = 4 ^ the preceding number [out-of-range of most computer
etc.

Let us write Fi(x) = x [i] x ; Indeed it will be more easy to
illustrate diagonalization on one variable function:

Thus F1(x) = x + x;  F2(x) = x * x, F3(x) = x ^ x, F4(x) = x [4] x,
F5(x) = x [5] x, F6(x) = x [6] x, etc.

This gives us an infinite list of one variable growing functions F0 F1,
F2, F3, F4, F5, F6, F7, ...

Please note that I could have taken Hal Finney list,  H0 H1 H2 H3 H4 H5
H6 H7 H8 H9 ...where H0(x) = factorial(x),
H1(x) =  factorial(factorial 2), H2(x) = factorial(factorial (factorial
(x))), ...

Mmmh... I am realizing it will even be easier to diagonalize
transfinitely with Hal Finney's functions than with the traditional
one, because with Hal Finney's one we will not been obliged of doing
some back and forth between one and two variable functions.

Anyway,  let

F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 ...

be your favorite sequence of one-variable more and more growing
function. (I recall all function here are function defined on N and
with value in N; where N = the set of natural numbers : 0, 1, 2, 3, ...

Here is a growing function, build from that class from diagonalization:

g(x) = Fx(x) + 1   (in english: to compute g(x), search the xth
function in your sequence, and apply it to x and then add 1.

For example  g(3) = F3(3) + 1, g(245) = F245(245) + 1, etc.

Exercises:

0) Could you evaluate roughly the number of digit of 4 [4] 4 ? What
about the number of digit of fact(fact(fact(fact 4

1) is the diagonal g function a growing function? Could g belong to the
initial sequence, does g grows more quickly than any function in the
initial sequence, and in what sense precisely.

2) Could you find a function, and even a new sequence of functions more
and more growing, and growing more than the function g?

3) Do you see why it is said that g is build by diagonalization? Where
is the diagonal?

4) Is there a universal sequence of growing functions, i. e. containing
all computable growing functions?

Must already go. Sorry for this quick piece. Solution tomorrow. Hope
before missing the real start ... Any comments , critics or suggestions
are welcome ...

Bruno

http://iridia.ulb.ac.be/~marchal/

I don't have time right now for detailed computations, but I'll give a

g is the same as my f(n,n,n)+1, and I already commented that f(n,m,n)
is a growing function, since f(N,m,n) is growing for fixed N.  So
clearly g is growing.  As I said about f(n,m,n), the degree or -ation
of g grows as n (or x) grows.  I recognize that adding 1 to make g is
the classical diagonalization move.  It makes g different from any Fi
in the sequence Fi, i=1,2,3,...  And in fact, since we add 1, rather
than subtract 1, g is larger than any Fi.

I'm having a problem with accepting g, or even my original f(n,n,n), as
a function in the same sense as with a fixed degree or -ation of
operation.  This is because the definition of the function changes
depending on the value taken in the domain of the function.  Is this
valid.

However, if we just ignore this problem, throwing caution to the wind,
then the next logical iteration of diagonalization is to do the
-ation thing on g and then diagonalize.  Let Gi(x) = g(x) [i] g(x),
then let h(x) = Gx(x) + 1.

Tom

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### Re: Smullyan Shmullyan, give me a real example



Le 22-mai-06, à 18:20, Tom Caylor a écrit :

Bruno Marchal wrote:
...
I give, for all, one last exercise before introducing diagonalization:
define recursively in an explicit way the operation [i+1] from the
preceding operation [i]. If you know a computer language (Fortran,
Lisp, Prolog, c++, Java, whatever ...) write the program. If you don't
know any such language, read my combinators posts and program those
function with S and K, (if you have the time). Well, just be sure you

Must leave now.

Bruno

I would have thought that my previous result captures this:

Generalizing this, given the function in the sequence corresponding
to
the operation of degree N.

f(N,m,n) = f(N-1,m,...f(N-1,m,n)...) (f(N-1) taken n times)

Well, if the representation is explicit, you should avoid the
But I think I see what you mean.

If we express my f(i,m,n) as your m [i] n, then this would be

m [i] n = m [i-1] (m [i-1] (...(m [i-1] n)...)   ( [i-1] taken n
times )

Same remark.

Or if we just look at m [i] m to keep it simpler as you suggest,

m [i] m = m [i-1] (m [i-1] (...(m [i-1] m)...)   ( [i-1] taken n
times )

I guess you mean m times   (not n).

In terms of a program, in a sort of pseudocode, to compute m [i] n,

initialize result to (m [i-1] n)
do the following n-1 times
set result to (m [i-1] result)
end do

All right, this is explicit.  Personally  I prefer recursive coding.
This is allowed in most modern language. But let us not take such
implementation issue too seriously.
The best would consist in implementing it by yourself on some real
machine, in case you would doubt your code.

The input is (m [i-1] n), the end result is m [i] n.  If we simply
want m [i] m, then set the input to (m [i-1] m).

Of course in a real computer language you would have to worry about
numerical representation and storage.

Many high level language make it possible  not to  worry about such
representations, and everything I will develop does not depend on those
issues. More latter.

Bruno

PS I got your three last emails  in double (in each post)  I don't know
why.

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example



Le 21-mai-06, à 10:53, Russell Standish a écrit :

On Thu, May 18, 2006 at 11:38:24AM +0200, Bruno Marchal wrote:

Also the universal dovetailer idea is also one of those that is
fairly
obvious, and might have been discovered a number of times
independently.

I'm not sure it is so easy, and in the present case I have never heard
Frankly I am not sure you got it right. I guess it is subtle: there is
a need of some amount in computer science to be astosnished that such
a
thing is logically possible. I will not develop this here because I
intend to make this clear in my reply (or sequence of replies) to Tom
and George.

I'm not sure why a knowledge of computer science would make the UD
astonishing. If anything, I would have thought the opposite.

In a sense, you are obviously right.  That is why I said some
knowledge of comp science or even just in math will make the existence
of the UD, and of the Universal Machine astonishing. Precisely it is
the knowledge of diagonalization. Godel will miss the universal machine
and Church thesis, and will describe those things as a sort of miracle.
More later.  I will comment again with much more detail the rest of
your post much later. If I comment it here now I will introduce
confusion. It is preferable people get much more familiarity with the
effective and not effective daigonalisations procedures before, I
think.

I'm

Thanks for telling,

The notion of dovetailing is really the theory behind timesharing, so
simple dovetailing must be pretty obvious, at least since the early
seventies.

That one can dovetail on all possible programs must be pretty obvious
once one realises that these can be enumerated. Of course the
philosophial consequences of being able to do this is not so obvious,
and as far as I know, you are the first person to have thought about
that.

Without the philosophical consequences, one would just think so
what? So it is perhaps not surprising noone mentioned the UD before
you.

Then I am showing that the appearances of persons and realities are
due to the incompleteness phenomena. I guess this is also a fairly
simple idea in the air, but, like the UD, I have not seen it develop
elsewhere, and it still gives me an hard and long time to make it
clear
as this very list can illustrate. And of course I can also be wrong,
also. My work mainly consists in making that idea testable (and
*partially* tested).

I sympathise, but I'm still having trouble getting the connection
too. Nevertheless, I find it intriguing.

Which connection? Is it not utterly obvious that, IF we are (hopefully
reasonable) machine, THEN we will learn something genuine by studying
possible neighborhoods and or histories?
What is not obvious, is that computer science put strong constraints on
the nature of possible machine realities, and with comp a case is made
that all constraints comes from number theoretical relations
(intensional and extensional(*)) and associated measures.
And then the comp hyp is used just for making things easy. The only
fundamental assumption which is needed for the reversal physics/numbers
is the hypothesis of correct self-reference. But I don't want to
anticipate at this stage.

Bruno

(*) extensional:  number represents themselves; intensional: number can
be used as code. Grosso modo: extensional number theory = number
theory; intensional number theory = computer science, information
science, provability logic.

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example


Le 23-mai-06, à 06:57, George Levy a écrit :

One can create faster and faster rising functions and larger and larger number until one is blue in the face. The point is that no matter how large a finite number n one defines, I can stand on the shoulder of giants and do better by citing n+1 using simple addition.

Now if somehow one came up with a finite number n so large that I am not allowed to say n+1 as if I was up against an overflow limitation similar to that found in computers, then there would be no physical way for me to invent or cite a larger number.  So it seems that if we are to define a largest finite number we must define it in conjunction with the number b of bits that we are allowed to use to express this number. For a given number of bits b the largest number would be n(b).

If we use the Ackerman series of functions we need 1 bit for addition, 2 bits for multiplication, 3 bits for exponentiation, 4 bits for tetration etc... These bits are required in addition to the bits for the input parameter(s) of the function.

What is the largest number of bits which are available to me to define an Ackerman function or some other fast rising function? Possibly the number of particles in the universe? I  don't know if the fairy would be satisfied or if I could personally herd all those bits.

The fairy gives you some amount of papers. She is fair enough to provide more if you ask politely ;)
The goal is to name a finite but as huge as possible number. It does not (obviously) consist in naming the biggest number (which does not exist as your little reasoning above shows clearly), nor does it consist in writing the best possible solution with respect to an available number of bits, (although we *will* arrive at this (much less simple) problem later).

Is she expecting me to hand in a piece of paper with the number written on it? Maybe then the answer would be the number generated by the largest Ackerman function that I can write with a very fine pen on this piece of paper.

Actually this can be considered as a good answer in the sense that the Ackermann number are already unimaginably gigantic, but that's nothing compared to the number which we will obtain by diagonalizations. Remember that my goal is to explain diagonalization. Actually, the goal of this thread is to explain Smullyan's heart of the matter in his FU book. For this we need not only to understand diagonalisation, but we will need to understand varieties of effective (programmable) and non effective diagonalizations before. I'm a bit sorry for the work I'm asking you ...

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example

 of addition, which is the preceding function in the
sequence of functions.

f(2,m,n) = f(2,m,n-1) + m
= f(1,m,f(2,m,n-1))
= f(1,m,f(2,m,n-2)+m)
= f(1,m,f(1,m,f(2,m,n-2)))
= f(1,m,f(1,m,f(2,n-3,m)+m))
= f(1,m,f(1,m,f(1,m,f(2,m,n-3
= f(1,m,...f(2,m,n-n)+m)...) (f(1) taken n-1 times,
f(2) taken 1 time)
= f(1,m,...f(2,m,n-n))  (f(1) taken n times,
f(2) taken 1 time)
= f(1,m,...f(2,m,0)) (f(1) taken n times,
f(2) taken 1 time)
= f(1,m,...f(1,m,0)) (f(1) taken n times)
= m * n

The above is a formal way of saying that multiplication of m and n is
adding n m's together.  We knew that.

Generalizing this, given the function in the sequence corresponding to
the operation of degree N.

f(N,m,n) = f(N-1,m,...f(N-1,m,n)) (f(N-1) taken n times)

The above is a formal way of saying that f(N)ing of m and n together is
the same as f(N-1)ing n m's together.

The sequence of ever growing functions is defined as f(N,m,n) for N=
1,2,3,...
Given a function of degree N, I take the growth of f(N,m,n) as defined
as its magnitude as n approaches infinity.

So here's a thought toward finding a function that's bigger than
any function in this sequence.  Define the following function.

d(m,n) = f(1,m,...f(n,m,n))

Note that n has been placed not only as the counting operation, but
also as the degree!  So now as n approaches infinity, the degree
approaches infinity. (!)  So here is a single function that has a
degree of operation that is higher than any function of a given degree
of operation.

Am I on the right track?

Tom

http://iridia.ulb.ac.be/~marchal/

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From: Bruno Marchal [EMAIL PROTECTED]
Subject: Re: Smullyan Shmullyan, give me a real example
Date: Mon, 22 May 2006 14:29:26 +0200
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Hi Tom,

Apparently you have (re)discover Ackermann function which indeed
provide formally a sequence of more and more growing functions, similar
to the sequence I was pointing too.
I will present it in a easier way for the benefit of the others, but
also for using a presentation which will facilitate the future
successive diagonalizations. In your second recent post you got the
first diagonalization right (or almost right) but the diag is quite
hidden and it would be hard to continue the process.

(Note that you could have define f(0, m, n) as the successor function)

presentation, the functions
f(1, m, n)is  m  +  n (or   m  [1]  n, writting   +
as [1]   ;  for first function)

f(2, m, n)is  m  *  n  (or m [2]  n)
f(3, m, n)is  m  ^  n  (or m [3] n)
f(4, m, n)is  m  [4]  n
f(5, m, n)is  m  [5]   n
etc

It will be easier to diagonalize functions of one argument, that is x +
x, x * x, x ^ x, x [4] x, x[5]x, etc.

Let us see their values on 10:

1) 10 + 10 = 20
2) 10 * 10 =  10+10+10+10+10+10+10+10+10 = 100  (ten sums)
3) 10 ^ 10 = 10*10*10*10*10*10*10*10*10 =  100  (ten products)
4) 10 [4] 10 = 10 ^ (10 ^ (10 ^ (10 ^ (10 ^ (10 ^ (10 ^ (10 ^  (10 ^
10) (ten exponentiations) Note the parenthesis:
schoolboys/girls know that a ^ (b ^ c) is in general different and
bigger than (a ^ b) ^ c .

Note also that 10 [4] 10 is already so big that the known observed part

### Re: Smullyan Shmullyan, give me a real example


Bruno Marchal wrote:
...
I give, for all, one last exercise before introducing diagonalization:
define recursively in an explicit way the operation [i+1] from the
preceding operation [i]. If you know a computer language (Fortran,
Lisp, Prolog, c++, Java, whatever ...) write the program. If you don't
know any such language, read my combinators posts and program those
function with S and K, (if you have the time). Well, just be sure you

Must leave now.

Bruno

I would have thought that my previous result captures this:

Generalizing this, given the function in the sequence corresponding to
the operation of degree N.

f(N,m,n) = f(N-1,m,...f(N-1,m,n)...) (f(N-1) taken n times)

If we express my f(i,m,n) as your m [i] n, then this would be

m [i] n = m [i-1] (m [i-1] (...(m [i-1] n)...)   ( [i-1] taken n
times )

Or if we just look at m [i] m to keep it simpler as you suggest,

m [i] m = m [i-1] (m [i-1] (...(m [i-1] m)...)   ( [i-1] taken n
times )

In terms of a program, in a sort of pseudocode, to compute m [i] n,

initialize result to (m [i-1] n)
do the following n-1 times
set result to (m [i-1] result)
end do

The input is (m [i-1] n), the end result is m [i] n.  If we simply
want m [i] m, then set the input to (m [i-1] m).

Of course in a real computer language you would have to worry about
numerical representation and storage.

Tom

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From: Tom Caylor [EMAIL PROTECTED]
Subject: Re: Smullyan Shmullyan, give me a real example
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Bruno Marchal wrote:
...
I give, for all, one last exercise before introducing diagonalization:
define recursively in an explicit way the operation [i+1] from the
preceding operation [i]. If you know a computer language (Fortran,
Lisp, Prolog, c++, Java, whatever ...) write the program. If you don't
know any such language, read my combinators posts and program those
function with S and K, (if you have the time). Well, just be sure you

Must leave now.

Bruno

I would have thought that my previous result captures this:

Generalizing this, given the function in the sequence corresponding to
the operation of degree N.

f(N,m,n) = f(N-1,m,...f(N-1,m,n)...) (f(N-1) taken n times)

If we express my f(i,m,n) as your m [i] n, then this would be

m [i] n = m [i-1] (m [i-1] (...(m [i-1] n)...)   ( [i-1] taken n
times )

Or if we just look at m [i] m to keep it simpler as you suggest,

m [i] m = m [i-1] (m [i-1] (...(m [i-1] m)...)   ( [i-1] taken n
times )

In terms of a program, in a sort of pseudocode, to compute m [i] n,

initialize result to (m [i-1] n)
do the following n-1 times
set result to (m [i-1] result)
end do

The input is (m [i-1] n), the end result is m [i] n.  If we simply
want m [i] m, then set the input to (m [i-1] m).

Of course in a real computer language you would have to worry about
numerical representation and storage.

Tom

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### Re: Smullyan Shmullyan, give me a real example


On Thu, May 18, 2006 at 11:38:24AM +0200, Bruno Marchal wrote:

Also the universal dovetailer idea is also one of those that is fairly
obvious, and might have been discovered a number of times
independently.

I'm not sure it is so easy, and in the present case I have never heard
Frankly I am not sure you got it right. I guess it is subtle: there is
a need of some amount in computer science to be astosnished that such a
thing is logically possible. I will not develop this here because I
intend to make this clear in my reply (or sequence of replies) to Tom
and George.

I'm not sure why a knowledge of computer science would make the UD
astonishing. If anything, I would have thought the opposite. I'm

The notion of dovetailing is really the theory behind timesharing, so
simple dovetailing must be pretty obvious, at least since the early
seventies.

That one can dovetail on all possible programs must be pretty obvious
once one realises that these can be enumerated. Of course the
philosophial consequences of being able to do this is not so obvious,
and as far as I know, you are the first person to have thought about that.

Without the philosophical consequences, one would just think so
what? So it is perhaps not surprising noone mentioned the UD before you.

Then I am showing that the appearances of persons and realities are
due to the incompleteness phenomena. I guess this is also a fairly
simple idea in the air, but, like the UD, I have not seen it develop
elsewhere, and it still gives me an hard and long time to make it clear
as this very list can illustrate. And of course I can also be wrong,
also. My work mainly consists in making that idea testable (and
*partially* tested).

I sympathise, but I'm still having trouble getting the connection
too. Nevertheless, I find it intriguing.

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example



One can create faster and faster rising functions and larger and larger
number until one is blue in the face. The point is that no matter how
large a finite number n one defines, I can stand on the
shoulder of giants and do better by citing n+1 using simple addition.

Now if somehow one came up with a finite number n so large that I
am not allowed to say n+1 as if I was up against an overflow
limitation similar to that found in computers, then there would be no
physical way for me to invent or cite a larger number. So it seems
that if we are to define a largest finite number we must define
it in conjunction with the number b of bits that we are allowed
to use to express this number. For a given number of bits b the largest
number would be n(b).

If we use the Ackerman series of functions we need 1 bit for addition,
2 bits for multiplication, 3 bits for exponentiation, 4 bits for
tetration etc... These bits are required in addition to the bits for
the input parameter(s) of the function.

What is the largest number of bits which are available to me to
define an Ackerman function or some other fast rising function?
Possibly the number of particles in the universe? I don't know if the
fairy would be satisfied or if I could personally herd all those bits.
Is she expecting me to hand in a piece of paper with the number written
on it? Maybe then the answer would be the number generated by the
largest Ackerman function that I can write with a very fine pen on this
piece of paper.

George

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### Re: Smullyan Shmullyan, give me a real example


I've been working on this off and on when I get a chance, even before
my first guess.  My version of this defines an operation as a
recursive function f(N,m,n), where N is the degree of the operation.
m is one of the operands.   n is the other operand, which is the
counting operand.  n is the number iterations that the recursive
function is evaluated.

I'll call addition the operation of degree 1. So addition can be
defined as follows.

Initial value:
f(1,m,0) = m

Recursion rule:
f(1,m,k) = f(1,m,k-1) + 1

So for a given n,
f(1,m,n) = f(1,...f(1,m,0) + 1,... + 1)   (f(1) taken n times)
= f(1,m,0) + 1 + ... + 1(1 added n times)
= f(1,m,0) + n
= m + n

Note that counting can be separately defined as the operation of degree
0, but this didn't add to the current argument.  Counting is also

Multiplication is defined in a similar manner, as the operation of
degree 2.

Initial value:
f(2,m,0) = 0

Recursion rule:
f(2,m,k) = f(2,m,k-1) + m

So for a given n,
f(2,m,n) = f(2,...f(2,m,0) + m,... + m)   (f(2) taken n times)
= f(2,m,0) + m + ... + m(m added n times)
= f(2,m,0) + m * n
= m * n

Exponentiation is the operation of degree 3.

f(3,m,0) = 1
f(3,m,k) = f(3,m,k-1) * m
f(3,m,n) = f(3,...f(3,m,0) * m,...* m)   (f(3) taken n times)
= f(3,m,0) * m * ... * m(m multiplied n times)
= f(3,m,0) * m ^ n
= m ^ n

Just for kicks, I tried to define hyper-nentiation as the operation
of degree 4, with operator symbol @.  I was interested in what
the initial value of this would be: the mth root of m.  Any further
than this gets too weird for me.

f(4,m,0) = m^(1/m)
f(4,m,1) = m
f(4,m,k) = f(4,m,k-1) ^ m
f(4,m,n) = f(4,...f(4,m,0) ^ m,...^ m)   (f(4) taken n times)
= m ^ m ^ ... ^ m(m exponentiated n times)
= m @ n

Looking closer specifically at multiplication, we see that it is
defined in terms of addition, which is the preceding function in the
sequence of functions.

f(2,m,n) = f(2,m,n-1) + m
= f(1,m,f(2,m,n-1))
= f(1,m,f(2,m,n-2)+m)
= f(1,m,f(1,m,f(2,m,n-2)))
= f(1,m,f(1,m,f(2,n-3,m)+m))
= f(1,m,f(1,m,f(1,m,f(2,m,n-3
= f(1,m,...f(2,m,n-n)+m)...) (f(1) taken n-1 times,
f(2) taken 1 time)
= f(1,m,...f(2,m,n-n))  (f(1) taken n times,
f(2) taken 1 time)
= f(1,m,...f(2,m,0)) (f(1) taken n times,
f(2) taken 1 time)
= f(1,m,...f(1,m,0)) (f(1) taken n times)
= m * n

The above is a formal way of saying that multiplication of m and n is
adding n m's together.  We knew that.

Generalizing this, given the function in the sequence corresponding to
the operation of degree N.

f(N,m,n) = f(N-1,m,...f(N-1,m,n)) (f(N-1) taken n times)

The above is a formal way of saying that f(N)ing of m and n together is
the same as f(N-1)ing n m's together.

The sequence of ever growing functions is defined as f(N,m,n) for N=
1,2,3,...
Given a function of degree N, I take the growth of f(N,m,n) as defined
as its magnitude as n approaches infinity.

So here's a thought toward finding a function that's bigger than
any function in this sequence.  Define the following function.

d(m,n) = f(1,m,...f(n,m,n))

Note that n has been placed not only as the counting operation, but
also as the degree!  So now as n approaches infinity, the degree
approaches infinity. (!)  So here is a single function that has a
degree of operation that is higher than any function of a given degree
of operation.

Am I on the right track?

Tom

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From: Tom Caylor [EMAIL PROTECTED]
Subject: Re: Smullyan Shmullyan, give me a real example
Date: Sun, 21 May 2006 00:08:25 -0700
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I've been working on this off and on when I get a chance, even before
my first guess.  My version of this defines an operation as a
recursive function f(N,m,n), where N

### Re: Smullyan Shmullyan, give me a real example


To be slightly more clear

d(m,n) = f(1,m,f(2,m,f(3,m,f(4,m,...f(n,m,n)...)

Note that the it's only the innermost function that has degree n.  To
simplify things, I suppose we could just consider f(n,m,n) by itself.
This has the same property that as n approaches infinity, the degree of
operation approaches infinity.  This gives a larger growth (as n
approaches infinity) than fixing the degree at any finite number.

And then, instead of substituting n into the degree, we could
substitute things like f(n,m,n) into the degree to get f(f(n,m,n),m,n).

Tom

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Subject: Re: Smullyan Shmullyan, give me a real example
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To be slightly more clear

d(m,n) = f(1,m,f(2,m,f(3,m,f(4,m,...f(n,m,n)...)

Note that the it's only the innermost function that has degree n.  To
simplify things, I suppose we could just consider f(n,m,n) by itself.
This has the same property that as n approaches infinity, the degree of
operation approaches infinity.  This gives a larger growth (as n
approaches infinity) than fixing the degree at any finite number.

And then, instead of substituting n into the degree, we could
substitute things like f(n,m,n) into the degree to get f(f(n,m,n),m,n).

Tom

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### Re: Smullyan Shmullyan, give me a real example


Le 19-mai-06, à 23:46, George Levy a écrit :

Bruno Marchal wrote:
Now I think I should train you with diagonalization. I give you an
exercise: write a program which, if executed, will stop on the biggest
possible natural number. Fairy tale version: you meet a fairy who
propose you a wish. You ask to be immortal but the fairy replies that
she has only finite power. So she can make you living as long as you
wish, but she asks precisely how long. It is up too you to describe
precisely how long you want to live by writing a program naming that
big (but finite) number. You have a limited amount of paper to write
your answer, but the fairy is kind enough to give you a little more if
You can ask the question to very little children. The cutest answer I
got was 7 + 7 + 7 + 7 + 7 (by a six year old). Why seven? It was the
age of his elder brother!

Hint: try to generate an infinite set S of more and more growing and
(computable) functions, and then try to diagonalize it. S can be
{addition, multiplication, exponentiation,   (?)}. More hints
and answers later. I let you think a little bit before. (Alas it looks
I will be more busy in may than I thought because my (math) students
want supplementary lessons this year ...).

Any potentially largest finite number n that I could name could be incremented by 1 so this finite number could not be the largest. The trick is not to name a particular number but to specify a method to reach the unreachable.

Well, if *you* try to give the biggest natural number, *you* will never stop, and you will not succeed in specifying a large number, and the fairy will not make your wish coming true, and you will gain nothing. It would be absurd not saying 10^100 under the pretext that you would have prefer to live (10^100)+1 years.
By trying to reach the unreachable, well, you are anticipating another fairy which I will present to you once you will be able to diagonalize without even thinking ...

Method 1) Use the fairy power against her.

I see you like to live dangerously. Or to die dangerously should I say

She says she has finite power. Ask for precisely the largest number of days she can provide with her finite power.

*any* FINITE number !
It is up to you to choose one in particular. And to succeed in describing which one.
Recall that the set of all finite things (or numbers) is infinite.

This method is similar to the robber's response when the victim asks him how much money do you want?: All the money in your pocket.

Method 2) Use the concept of limits Ask for as many days it would take to obtain a sum of 2 as terms in the series 1+1/2 + 1/4 + 1/8 + 1/16. If the fairies knows any math she may argue that the series never reaches 2.

And she is right!

On the other hand I may argue that in the limit it does reach 2.

Yes but to reach that limit you need an infinity of additions. The serie is convergent, and you can go as close as you want to 2 in finite steps, but 2 itself requires an infinity of steps, and the fairy asks you to specify a precise number, actually the biggest you can specify precisely (through some algorithm, program, description) using a reasonable amount of paper. The better is to describe a growing function applied to some number. Although it looks a little bit ridiculous, the child solution: 9 + 9 contains the basic good idea: applying a growing function you know (or can program), like +  on a big number that you know, like 9. Of course 9 * 9  is better, and 9^9 is still better, ... (?)

Method 3) Come up with a unprovably non-halting problem:

Again, you are anticipating on the next sort of fairy I will present later. The current fairy is somehow constructivist, and ask you to specify a number as big as you can describe, but it must be a precisely well defined  number.

For example ask for as many days as required digits in PI to prove that PI has a single repetition of a form such that digits 1 to n match digits  n+1 to 2n. For example 2^0.5 = 1.4142135... has a  single repetition (1 4 match 1 4) in which digits 1 to 2 match digits 3 to 4. Similarly  79^0.5=8.8881944 and 147^0.5= 12.12435565. Note that the repetition must include all numbers 1 to n from the beginning and match all number n+1 to 2n The problem with this approach is I don't know for sure if PI is repeatable or non-repeatable (according to above requirements.)  I don't even know if this problem is unprovable. All I know is that the probability for any irrational to have a single repeat is about 0.. For PI the probability is much lower since I already know PI to a large number of digits and as far as I can see it does not repeat. However, with this approach I could be taking chances.

Indeed.

Diagonalization clearly allows you to specify a number outside any given set of number, but I have not been able to weave it into this argument.

I will say more in my reply to Hal Finney who gives a good start.



### Re: Smullyan Shmullyan, give me a real example



Le 20-mai-06, à 01:17, Hal Finney a écrit :

Bruno writes:
Meanwhile just a few questions to help me. They are hints for the=20
problem too. Are you familiar with the following recursive
program=20
for computing the factorial function?

fact(0) = 1
fact (n) = n * fact(n - 1)

Could you compute fact 5, from that program? Could you find a
similar=20
recursive definition (program) for multiplication (assuming your=20
Could you define exponentiation from multiplication in a similar way?
=20
Could you find a function which would grow more quickly than=20
exponentiation and which would be defined from exponentiation like=20
exponentiation is defined from multiplication? Could you generalize
all=20
this and define a sequence of more and more growing functions.
Could=20
you then diagonalise effectively (=3D writing a program who does
the=20
diagonalization) that sequence of growing functions so as to get a=20
function which grows more quickly than any such one in the
preceding=20
sequence?

Here's what I think you are getting at with the fairy problem.  The
point
is not to write down the last natural number, because of course there
is no such number.

Right!

Rather, you want to write a program which represents
(i.e. would compute) some specific large number, and you want to come
up
with the best program you can for this, i.e. the program that produces
the largest number from among all the programs you can think of.

... and write on a reasonable amount of paper provided by the fairy. Of
course it must be a finite description.

If we start with factorial, we could define a function func0 as:

func0(n) = fact(n)

Now this gets big pretty fast.  func0(100) is already enormous, it's
like
a 150 digit number.  However we can stack this function by calling
it on itself.  func0(func0(100)) is beyond comprehension.  And we can
generalize, to call it on itself as many times as we want, like n
times:

func1(n) = func0(func0(func0( ... (n))) ... )))

where we have nested calls of func0 on itself n times.

All right. You provide a sequence of growing functions: func0(n),
func0(func0(n), etc. And then you get func1(n) by an n-iteration of
func0.

This really gets
bigger fast, much faster than func0.

Then we can nest func1:

func2(n) = func1(func1(func1( ... (n))) ... )))

where again we have nested calls of func1 on itself n times.  We know
that func1(n) gets bigger so fast, func1(func1(n)) will get bigger
amazingly faster, and of course with n of them it is that much faster
yet.

This clearly generalizes to func3, func4,

Now we can step up a level and define hfunc1(n) = funcn(n), the nth
function along the path from func1, func2, func3,   Wow, imagine
how fast that gets bigger.  hfunc is for hyperfunc.

Then we can stack the hfuncs, and go to an ifunc, a jfunc, etc.  Well,
my terminology is getting in the way since I used letters instead of
numbers here.  But if I were more careful I think it would be possible
to continue this process more or less indefinitely.  You'd have program
P1 which continues this process of stacking and generalizing, stacking
and generalizing.  Then you could define program P2 which runs P1
through
n stack-and-generalize sequences.  Then we stack-and-generalize P2,
etc.
It never ends.  But it's not clear to me how to describe the process
formally.

All right.  I will do this a little bit more formally, but you got the
right idea. Eventually we will see that there is no systematic way to
get the biggest number by such a procedure, so that it is not so
important which one to choose, except it is better to choose one which
is such that we can easily describe a big part of that sequence of
sequences of sequences ... of growing functions.
Your stack and generalize seems to correspond to diagonalizations.

So we have this ongoing process where we define a series of functions
that get big faster and faster than the ones before.  I'm not sure how
we
use it.  Maybe at some point we just tell the fairy, okay, let me live
P1000(1000) years.

Yes. Well, if you have still some remaining place you can write
P1000(P1000(1000)).

That's a number so big that from our perspective it
seems like it's practically infinite.

Absolutely!  We should ask the fairy if she provides the growing
brain needed for living a so long time. If the brain is not growing,
given that its dimension are very small compare to such big number,
it will cycle!

But of course from the infinite
perspective it seems like it's practically zero.

Right, but then this will be the case with any fairy capable of
providing only finite (but long) life. In a sense, all numbers are big,
and even *very big*, and very very big, etc ... Except for a finite
number of them.

Computer scientist say almost all number have property P when all
numbers have 

### Re: Smullyan Shmullyan, give me a real example



Bruno Marchal wrote:

Now I think I should train you with diagonalization. I give you an
exercise: write a program which, if executed, will stop on the biggest
possible natural number. Fairy tale version: you meet a fairy who
propose you a wish. You ask to be immortal but the fairy replies that
she has only finite power. So she can make you living as long as you
wish, but she asks precisely how long. It is up too you to describe
precisely how long you want to live by writing a program naming that
big (but finite) number. You have a limited amount of paper to write
your answer, but the fairy is kind enough to give you a little more if
You can ask the question to very little children. The cutest answer I
got was "7 + 7 + 7 + 7 + 7" (by a six year old). Why seven? It was the
age of his elder brother!

Hint: try to generate an infinite set S of more and more growing and
(computable) functions, and then try to diagonalize it. S can be
{addition, multiplication, exponentiation,   (?)}. More hints
and answers later. I let you think a little bit before. (Alas it looks
I will be more busy in may than I thought because my (math) students
want supplementary lessons this year ...).

Any potentially largest finite number n that I could name could be
incremented by 1 so this finite number could not be the largest. The
trick is not to name a particular number but to specify a method to
reach the unreachable.

Method 1) Use the fairy power against her. She says she has "finite
power". Ask for precisely the largest number of days she can provide
with her "finite power." This method is similar to the robber's
response when the victim asks him "how much money do you want?": "All

Method 2) Use the concept of "limits" Ask for as many days it would
take to obtain a sum of 2 as terms in the series 1+1/2 + 1/4 + 1/8 +
1/16. If the fairies knows any math she may argue that the series
never reaches 2. On the other hand I may argue that "in the limit" it
does reach 2.

Method 3) Come up with a unprovably non-halting problem: For
example ask for as many days as required digits in PI to prove that PI
has a single repetition of a form such that digits 1 to n match
digits n+1 to 2n. For example 2^0.5 = 1.4142135... has a single
repetition (1 4 match 1 4) in which digits 1 to 2 match digits 3 to 4.
Similarly 79^0.5=8.8881944 and 147^0.5= 12.12435565. Note that the
repetition must include all numbers 1 to n from the beginning and match
all number n+1 to 2n The problem with this approach is I don't know for
sure if PI is repeatable or non-repeatable (according to above
requirements.) I don't even know if this problem is unprovable. All I
know is that the probability for any irrational to have a single repeat
is about 0.. For PI the probability is much lower since I already
know PI to a large number of digits and as far as I can see it does not
repeat. However, with this approach I could be taking chances.

Diagonalization clearly allows you to specify a number outside any
given set of number, but I have not been able to weave it into this
argument.

George

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### Re: Smullyan Shmullyan, give me a real example


Bruno writes:
Meanwhile just a few questions to help me. They are hints for the=20
problem too. Are you familiar with the following recursive program=20
for computing the factorial function?

fact(0) =3D 1
fact (n) =3D n * fact(n - 1)

Could you compute fact 5, from that program? Could you find a similar=20
recursive definition (program) for multiplication (assuming your=20
Could you define exponentiation from multiplication in a similar way? =20
Could you find a function which would grow more quickly than=20
exponentiation and which would be defined from exponentiation like=20
exponentiation is defined from multiplication? Could you generalize all=20
this and define a sequence of more and more growing functions. Could=20
you then diagonalise effectively (=3D writing a program who does the=20
diagonalization) that sequence of growing functions so as to get a=20
function which grows more quickly than any such one in the preceding=20
sequence?

Here's what I think you are getting at with the fairy problem.  The point
is not to write down the last natural number, because of course there
is no such number.  Rather, you want to write a program which represents
(i.e. would compute) some specific large number, and you want to come up
with the best program you can for this, i.e. the program that produces
the largest number from among all the programs you can think of.

If we start with factorial, we could define a function func0 as:

func0(n) = fact(n)

Now this gets big pretty fast.  func0(100) is already enormous, it's like
a 150 digit number.  However we can stack this function by calling
it on itself.  func0(func0(100)) is beyond comprehension.  And we can
generalize, to call it on itself as many times as we want, like n times:

func1(n) = func0(func0(func0( ... (n))) ... )))

where we have nested calls of func0 on itself n times.  This really gets
bigger fast, much faster than func0.

Then we can nest func1:

func2(n) = func1(func1(func1( ... (n))) ... )))

where again we have nested calls of func1 on itself n times.  We know
that func1(n) gets bigger so fast, func1(func1(n)) will get bigger
amazingly faster, and of course with n of them it is that much faster yet.

This clearly generalizes to func3, func4,

Now we can step up a level and define hfunc1(n) = funcn(n), the nth
function along the path from func1, func2, func3,   Wow, imagine
how fast that gets bigger.  hfunc is for hyperfunc.

Then we can stack the hfuncs, and go to an ifunc, a jfunc, etc.  Well,
my terminology is getting in the way since I used letters instead of
numbers here.  But if I were more careful I think it would be possible
to continue this process more or less indefinitely.  You'd have program
P1 which continues this process of stacking and generalizing, stacking
and generalizing.  Then you could define program P2 which runs P1 through
n stack-and-generalize sequences.  Then we stack-and-generalize P2, etc.
It never ends.  But it's not clear to me how to describe the process
formally.

So we have this ongoing process where we define a series of functions
that get big faster and faster than the ones before.  I'm not sure how we
use it.  Maybe at some point we just tell the fairy, okay, let me live
P1000(1000) years.  That's a number so big that from our perspective it
seems like it's practically infinite.  But of course from the infinite
perspective it seems like it's practically zero.

Hal Finney

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### Re: Smullyan Shmullyan, give me a real example



Le 11-mai-06, à 13:38, Russell Standish a écrit :

On Thu, May 11, 2006 at 01:00:31PM +0200, Bruno Marchal wrote:

I think Schroedinger used the cat for explaining a paradoxical feature
of QM, and I have not see suggestions by him that comp leads to either
many world or quantum immortality (as Everett and Deutsch will do for
the many-world, but not the immortality question though.
I think that in the priority matter (a boring subject but then
friends said that I must defend myself a little bit more) the criteria
is the date of the publication. It is one thing to get an idea and a
different thing to publish it. You need to fçind the idea but also to

James Higgo found a 1986 publication by Euan Sqires that mentions the
immortality argument. Perhaps that's not too much earlier for you to
claim independent discovery in your 1988 paper. Still the point is,
its one of those ideas that's floating around anyway - in the ether,
so to speak.

Sure.

Also the universal dovetailer idea is also one of those that is fairly
obvious, and might have been discovered a number of times
independently.

I'm not sure it is so easy, and in the present case I have never heard
Frankly I am not sure you got it right. I guess it is subtle: there is
a need of some amount in computer science to be astosnished that such a
thing is logically possible. I will not develop this here because I
intend to make this clear in my reply (or sequence of replies) to Tom
and George.

In some ways, these ideas are too simple for the issue of priority to
be taken seriously. Perhaps, but the fame game is fickle
indeed. Famous people are often not famous for their most important
work. My most cited paper according to Google Scholar On complexity
and
emergence doesn't contain any original ideas at all! (Its a digestion
of what I've read on the topics)

On the other hand your COMP ontological reversal idea is truly
unique. Hopefully you are right, and it goes down in history as your
greatest contribution to human knowledge.

Well thanks for that. In my opinion the UD, the UDA, the Universal
Machine and Church Thesis are all deeply linked together (once in the
TOE context).

About the priority,  I don't care so much, my point consisted mainly in
the fact that the quantum immortality is a sub-case of comp
immortality, which, by the way, can even be considered as a sub-case of
the usual Pythagorico-Platonist-Plotino-Cartesian  argument for the
immortality of the soul which has been proposed by the intellectual
greeks for about a millennium in Occident (more or less -500 to +500 JC
era).

Then I am showing that the appearances of persons and realities are
due to the incompleteness phenomena. I guess this is also a fairly
simple idea in the air, but, like the UD, I have not seen it develop
elsewhere, and it still gives me an hard and long time to make it clear
as this very list can illustrate. And of course I can also be wrong,
also. My work mainly consists in making that idea testable (and
*partially* tested).

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example



Le 16-mai-06, à 02:22, Russell Standish a écrit :

An observer attaches a meaning to the data e observes. The set of all
such meanings is semantic space or meaning space. I believe this is
necessarily a discrete set (but not necessarily finite).

If you have the time to define formally your meaning space I could be
interested. I don't founf it in your writings.

The details, of course are in my paper Why Occams Razor. To
summarise, an observer induces a map O(x) from the space of
descriptions, which is equivalent AFAIK to the output of your UD,

?   The UD has neither inputs nor outputs. (like any universe or
everything, note)

Perhaps I'm being a little casual in my terminology. What I'm
referring to is UD*.

But UD* is not even a program. It is the trace (at some level) of the
entire (infinite) execution of the UD.

Why does constructibility, or
otherwise have anything to do with the 1/3 person distinction?

It is the logic of the self-extending self. It is akin to Brouwer's
theory of consciousness, which is a root of its intuitionist
philosophy. It is among the confirmed point through the fact that the
theaetetical variant Bp  p (the soul hypostase), although not
constructive per se, does lead to an arithmetical interpretation of
intuitionism, like the intelligible and sensible matter hypostases
lead to a form of arithmetical quantization.

I am willing to concede that there is possibly more to the WR problem,
but I have yet to see it expressed in a manner I can understand :).

It is the whole point of the UDA to help making this clear. Perhaps I
am wrong but I think you underestimate the fact that the first person
are not aware of the delays (numbers of steps) make by the UD to reach
computational continuing states. Think about a highly discontinuous
function with continuous derivatives: we can only be conscious of the
derivative because we does not feel either the splitting or
bifurcation, nor the discontinuous jumps.

Everyone is free to download my last presentation of the UDA (my SANE
paper), and tell me at which point they believed the argument does not
proceed. A step could be wrong or not well supported perhaps, but until
now, honest scientists who take time to verify the argument does not
see anything wrong, and most abandon comp in a way or another because
they cannot really swallow the platonist reversal ...
Well, in the worst case I will come back next millennium ;-)

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example



Le 16-mai-06, à 17:31, Tom Caylor a écrit :

Bruno Marchal wrote:

Now I think I should train you with diagonalization. I give you an
exercise: write a program which, if executed, will stop on the biggest
possible natural number. Fairy tale version: you meet a fairy who
propose you a wish. You ask to be immortal but the fairy replies that
she has only finite power. So she can make you living as long as you
wish, but she asks precisely how long. It is up too you to describe
precisely how long you want to live by writing a program naming that
big (but finite) number. You have a limited amount of paper to write
your answer, but the fairy is kind enough to give you a little more if
You can ask the question to very little children. The cutest answer I
got was 7 + 7 + 7 + 7 + 7 (by a six year old). Why seven? It was the
age of his elder brother!

Hint: try to generate an infinite set S of more and more growing and
(computable) functions, and then try to diagonalize it. S can be
{addition, multiplication, exponentiation,   (?)}. More hints
and answers later. I let you think a little bit before. (Alas it looks
I will be more busy in may than I thought because my (math) students
want supplementary lessons this year ...).

Hope this can help; feel free to make *any* comments.

Remember that if all this is too technical, you can also just read
Plotinus and the (neo)platonist which, accepting comp or weaker form
of
Pythagorism,  do have a tremendous advance on most materialist of
today
... I think it could even provide more light on the practical death
issue. The role of G and G* is just to get the math correct for some
notion of quantifying the 1-person probabilities.

Bruno

(*)SANE paper html:
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.htm
SANE paper pdf:
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf

http://iridia.ulb.ac.be/~marchal/

In keeping with the incremental interactive process, here is a first
guess.  You simply start naming off the natural numbers in order.
After naming each number you say, That's not the largest possible
natural number, or That's not how long I want to live.  This
statement seems to play the role of diagonalization.

But it is not a finite process. The fairy asks you to give a well
defined number, in a finite time.

The process I've
just described can be defined with a finite number of symbols (I just
did it).  Thus, in a way you can say I've just named the largest
natural number.

You have just given a procedure for building a bigger number from any
number. The function which send n on n+1  does that trick. But the
fairy asks you for a number, not a function.

First question: Is this the same as Douglas Hoftstadter's supernatural
numbers (in his book Godel, Escher, Bach)?

I have read that quite good book, but I don't have it under the hand,
and I don't think the big number problem is related to its supernatural
numbers.

It seems the only way to
really understand his book is to read it cover-to-cover (because of all
the acronyms and his defining ideas with stories, etc.).  I wish I
would have read it cover-to-cover when I was young and had lots of time
on my hands (and lots of spare brain cells) or may I can just start
(quality) time as I need it.

Hofstadter wrote a good book, yes, but on the pedagogical side it does
not help so much by diluting the proof of Godel's theorem in many
interesting themes (Bach, Escher, AI, etc.).

Second question:  When we switch over from natural numbers to length
of life, it seems we need to specify units of time in order for the
specification of length of life to have any meaning.

You are right. Let us take *years.

This crosses us
over into the realm of meaning.  Length of life has no meaning apart
from an assignment of meaning or quality to the events that make up
life.  There seems to be some kind of diagonalization going on here (or
perhaps transcendence, independent from any diagonalization argument).
What good is MWI immortality (or any kind of immortality) if the
infinite sum of (units of time) * (quality or meaning) adds up to some
finite number?  Is it really immortality? Life is more than existence.

In the big number problem, immortality is not proposed by the fairy,
what is proposed is just a long but finite life.
Here too the quality is important. To stay a very long time awake in a
coffin is not pleasant. Also, to stay alive for a very long period
makes almost no sense if your brain is limited in space (bounded finite
machine eventually cycle when running a long time. Do you see why?).

The big number problem has been tackled by Archimedes. He got the
number 10^63. This is remarkable if you recall the very bad notation
for number used at that time. Today 10^63, although very big (the
universe seems to 

### Re: Smullyan Shmullyan, give me a real example


On Mon, May 15, 2006 at 03:51:56PM +0200, Bruno Marchal wrote:

Le 15-mai-06, à 13:59, Russell Standish a écrit :

OK, why not taking that difference [description/computation] into
account. I think it is a
crucial point.

I do :). However, its makes no difference as far as I can tell to the
Occam's razor issue.

You do? See below.

given a reference Turing machine U. This appears
to be a 3rd person description, but it need not be so.

I am not sure I understand.

Do you mean you don't think it is a 3rd person description, or do you
mean you don't think it can be anything else?

I  think it is a third person description.

That's what I suspect most people think. My point is that it needn't
be, and it is in fact inherently first person. I make this point in
many different papers, as well as my book.

In the fairness of scientific discussion, I am willing to be shown
wrong, of course :)

The details, of course are in my paper Why Occams Razor. To
summarise, an observer induces a map O(x) from the space of
descriptions, which is equivalent AFAIK to the output of your UD,

?   The UD has neither inputs nor outputs. (like any universe or
everything, note)

Perhaps I'm being a little casual in my terminology. What I'm
referring to is UD*.

to
the space of meanings.

Which space is it?  What do you mean (here) by meanings?

An observer attaches a meaning to the data e observes. The set of all
such meanings is semantic space or meaning space. I believe this is
necessarily a discrete set (but not necessarily finite).

If it is a
mathematical semantics then which one, if not, I don't understand. I

For any given meaning y, let omega(y,l) be the
number of equivalent descriptions of length l mapping to y (for
infinite length description we need the length l prefixes). So

omega(y,l) = |{x: O(x)=y  len(x)=l}|

Now P(y) = lim_{l-\infty} omega(y,l)/2^l is a probability
distribution, related to the Solomonoff-Levin universal
distribution.

C(y)=-log_2 P(y)

is a complexity measure related to Kolmogorov Complexity.

Note that this approach is non constructive (and thus cannot be first
person, at least as I use it and modelize it). I have already argued
that it can be refined through the notion of depth (a la Bennett),
which takes a notion of long computation into account; but it is
still incomplete relatively to the first person indeterminacy problem
(pertaining on the set of *all* (relative) computations, and not at all
on the set of descriptions).
The non-constructibility is a problem here, given the goal of deducing
physical laws or principles without physics.

And now I don't understand you. Why does constructibility, or
otherwise have anything to do with the 1/3 person distinction?

If you have succeed in eliminating all the many person pov - white
rabbits,  then publish!

Well, I have! One thing you can't accuse me of is not publishing my
ideas.

Frankly it seems to me you don't really address the first person issue
(and thus the mind/body issue).

Yes - you've said that before, and its a point I've never understood.

For example, what is your theory of
mind? In particular, do you say yes to the comp doctor?

Pretty much everything thing I've done summarises the theory of the
mind by the function O(x). It maps descriptions (aka bitstrings) to
meaning. I do make use of a robustness property, which essentially is
that O^{-1}(y) is not of measure zero, but that is about it.

In particular, none of my results depend on whether I would say yes to
the comp doctor or not.

I think that eventually, we have to limit ourself to the discourses
that a self-referentially correct machine (or entity, or growing
entities of such lobian kind) can have about herself and her
possibilities.

And I think you could be right, or even approximately right. At this
stage, we need to explore.

I am not saying your argument is wrong, just that is incomplete (and
unclear, but this could be my incompetence).

Bruno

Well, of course it is incomplete if you're looking for a TOE. For the
White Rabbit issue, the argument is quite simple. I have conceived of
the White Rabbit problem in a certain way: the unreasonable
effectiveness of mathematics, the (non-)failure of induction. It
certainly appears to me that the argument addresses this conclusively,
from a first person point of view, however, there is always room for
doubt that I have overlooked some nuance.

I am willing to concede that there is possibly more to the WR problem,
but I have yet to see it expressed in a manner I can understand :).

Where I suspect most people might come unstuck is justifying why
formula (1) from On Complexity and Emergence should be called
complexity. The reason comes down its connection with Kolmogorov
complexity - it is the 

### Re: Smullyan Shmullyan, give me a real example


Bruno Marchal wrote:

Now I think I should train you with diagonalization. I give you an
exercise: write a program which, if executed, will stop on the biggest
possible natural number. Fairy tale version: you meet a fairy who
propose you a wish. You ask to be immortal but the fairy replies that
she has only finite power. So she can make you living as long as you
wish, but she asks precisely how long. It is up too you to describe
precisely how long you want to live by writing a program naming that
big (but finite) number. You have a limited amount of paper to write
your answer, but the fairy is kind enough to give you a little more if
You can ask the question to very little children. The cutest answer I
got was 7 + 7 + 7 + 7 + 7 (by a six year old). Why seven? It was the
age of his elder brother!

Hint: try to generate an infinite set S of more and more growing and
(computable) functions, and then try to diagonalize it. S can be
{addition, multiplication, exponentiation,   (?)}. More hints
and answers later. I let you think a little bit before. (Alas it looks
I will be more busy in may than I thought because my (math) students
want supplementary lessons this year ...).

Hope this can help; feel free to make *any* comments.

Remember that if all this is too technical, you can also just read
Plotinus and the (neo)platonist which, accepting comp or weaker form of
Pythagorism,  do have a tremendous advance on most materialist of today
... I think it could even provide more light on the practical death
issue. The role of G and G* is just to get the math correct for some
notion of quantifying the 1-person probabilities.

Bruno

(*)SANE paper html:
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.htm
SANE paper pdf:
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf

http://iridia.ulb.ac.be/~marchal/

In keeping with the incremental interactive process, here is a first
guess.  You simply start naming off the natural numbers in order.
After naming each number you say, That's not the largest possible
natural number, or That's not how long I want to live.  This
statement seems to play the role of diagonalization.  The process I've
just described can be defined with a finite number of symbols (I just
did it).  Thus, in a way you can say I've just named the largest
natural number.

First question: Is this the same as Douglas Hoftstadter's supernatural
numbers (in his book Godel, Escher, Bach)?  It seems the only way to
really understand his book is to read it cover-to-cover (because of all
the acronyms and his defining ideas with stories, etc.).  I wish I
would have read it cover-to-cover when I was young and had lots of time
on my hands (and lots of spare brain cells) or may I can just start
(quality) time as I need it.

Second question:  When we switch over from natural numbers to length
of life, it seems we need to specify units of time in order for the
specification of length of life to have any meaning.  This crosses us
over into the realm of meaning.  Length of life has no meaning apart
from an assignment of meaning or quality to the events that make up
life.  There seems to be some kind of diagonalization going on here (or
perhaps transcendence, independent from any diagonalization argument).
What good is MWI immortality (or any kind of immortality) if the
infinite sum of (units of time) * (quality or meaning) adds up to some
finite number?  Is it really immortality? Life is more than existence.

Tom

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### Re: Smullyan Shmullyan, give me a real example



Le 15-mai-06, à 02:04, Russell Standish a écrit :

I guess it is a delicate point, a key point though, which overlaps the
ASSA/RSSA distinction (that is: the Absolute Self Sampling Assumption
versus the Relative Self Sampling Assumption).

If you identify a conscious first person history with a third
person
describable computation, it can be argued that an explanation for
physics can be given by Bayesian sort of anthropic reasoning based on
some  universal  probability distribution like Hall Finney's
Kolmogorovian UDist. Note tat this approach relies also on Church
Thesis. Here somehow the TOE will be a winning little program. I agree
that this would hunt away the third person white rabbits.

I disagree. The UDist comes from looking at the measure induced on a
set of descriptions

OK.

It is not the same. It changes the whole problem, especially from the
Relative SSA (Self-sampling assumption).

although the two
are not equivalent),

OK, why not taking that difference into account. I think it is a
crucial point.

given a reference Turing machine U. This appears
to be a 3rd person description, but it need not be so.

I am not sure I understand.

As I have
pointed out (but suspect it hasn't really sunk in yet), U can be
taken to be the observer erself.

I could agree, but U cannot *know* e is U. Need some bet or act of
faith.
In general if U describes the observer, he is a big number in need of
an explanation. I mean, the existence of  big stable U is what we try
to explain.

When done this way, there is a 1st
person universal distribution, with a corresponding 1st person Occam
razor theorem. And this implies the absence of 1st person white
rabbits.

I really don't understand.

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example


On Mon, May 15, 2006 at 11:17:35AM +0200, Bruno Marchal wrote:

Le 15-mai-06, à 02:04, Russell Standish a écrit :

I guess it is a delicate point, a key point though, which overlaps the
ASSA/RSSA distinction (that is: the Absolute Self Sampling Assumption
versus the Relative Self Sampling Assumption).

If you identify a conscious first person history with a third
person
describable computation, it can be argued that an explanation for
physics can be given by Bayesian sort of anthropic reasoning based on
some  universal  probability distribution like Hall Finney's
Kolmogorovian UDist. Note tat this approach relies also on Church
Thesis. Here somehow the TOE will be a winning little program. I agree
that this would hunt away the third person white rabbits.

I disagree. The UDist comes from looking at the measure induced on a
set of descriptions

OK.

It is not the same. It changes the whole problem, especially from the
Relative SSA (Self-sampling assumption).

although the two
are not equivalent),

OK, why not taking that difference into account. I think it is a
crucial point.

I do :). However, its makes no difference as far as I can tell to the
Occam's razor issue.

given a reference Turing machine U. This appears
to be a 3rd person description, but it need not be so.

I am not sure I understand.

Do you mean you don't think it is a 3rd person description, or do you
mean you don't think it can be anything else?

As I have
pointed out (but suspect it hasn't really sunk in yet), U can be
taken to be the observer erself.

I could agree, but U cannot *know* e is U. Need some bet or act of
faith.
In general if U describes the observer, he is a big number in need of
an explanation. I mean, the existence of  big stable U is what we try
to explain.

Sure. In fact the argument does not even hinge upon the observer being a
UTM. However, if not, then the distribution is not exactly universal
(hence my quotes below).

When done this way, there is a 1st
person universal distribution, with a corresponding 1st person Occam
razor theorem. And this implies the absence of 1st person white
rabbits.

I really don't understand.

Bruno

The details, of course are in my paper Why Occams Razor. To
summarise, an observer induces a map O(x) from the space of
descriptions, which is equivalent AFAIK to the output of your UD, to
the space of meanings. For any given meaning y, let omega(y,l) be the
number of equivalent descriptions of length l mapping to y (for
infinite length description we need the length l prefixes). So

omega(y,l) = |{x: O(x)=y  len(x)=l}|

Now P(y) = lim_{l-\infty} omega(y,l)/2^l is a probability
distribution, related to the Solomonoff-Levin universal
distribution.

C(y)=-log_2 P(y)

is a complexity measure related to Kolmogorov Complexity.

Basically this is an Occams Razor theorem - the probability of
observing something decreases dramatically with its observed
complexity. And this is a pure 1st person result. It doesn't get rid
of all white rabbits, but the remaining ones are dealt with the
Malcolm-Standish argument.

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example



Le 15-mai-06, à 13:59, Russell Standish a écrit :

OK, why not taking that difference [description/computation] into
account. I think it is a
crucial point.

I do :). However, its makes no difference as far as I can tell to the
Occam's razor issue.

You do? See below.

given a reference Turing machine U. This appears
to be a 3rd person description, but it need not be so.

I am not sure I understand.

Do you mean you don't think it is a 3rd person description, or do you
mean you don't think it can be anything else?

I  think it is a third person description.

snip

I really don't understand.

Bruno

The details, of course are in my paper Why Occams Razor. To
summarise, an observer induces a map O(x) from the space of
descriptions, which is equivalent AFAIK to the output of your UD,

?   The UD has neither inputs nor outputs. (like any universe or
everything, note)

to
the space of meanings.

Which space is it?  What do you mean (here) by meanings?  If it is a
mathematical semantics then which one, if not, I don't understand. I

For any given meaning y, let omega(y,l) be the
number of equivalent descriptions of length l mapping to y (for
infinite length description we need the length l prefixes). So

omega(y,l) = |{x: O(x)=y  len(x)=l}|

Now P(y) = lim_{l-\infty} omega(y,l)/2^l is a probability
distribution, related to the Solomonoff-Levin universal
distribution.

C(y)=-log_2 P(y)

is a complexity measure related to Kolmogorov Complexity.

Note that this approach is non constructive (and thus cannot be first
person, at least as I use it and modelize it). I have already argued
that it can be refined through the notion of depth (a la Bennett),
which takes a notion of long computation into account; but it is
still incomplete relatively to the first person indeterminacy problem
(pertaining on the set of *all* (relative) computations, and not at all
on the set of descriptions).
The non-constructibility is a problem here, given the goal of deducing
physical laws or principles without physics.

Basically this is an Occams Razor theorem - the probability of
observing something decreases dramatically with its observed
complexity. And this is a pure 1st person result.

?

It doesn't get rid
of all white rabbits, but the remaining ones are dealt with the
Malcolm-Standish argument.

If you have succeed in eliminating all the many person pov - white
rabbits,  then publish!

Frankly it seems to me you don't really address the first person issue
(and thus the mind/body issue). For example, what is your theory of
mind? In particular, do you say yes to the comp doctor?
I think that eventually, we have to limit ourself to the discourses
that a self-referentially correct machine (or entity, or growing
entities of such lobian kind) can have about herself and her
possibilities.

I am not saying your argument is wrong, just that is incomplete (and
unclear, but this could be my incompetence).

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example


On Thu, May 11, 2006 at 01:00:31PM +0200, Bruno Marchal wrote:

I think Schroedinger used the cat for explaining a paradoxical feature
of QM, and I have not see suggestions by him that comp leads to either
many world or quantum immortality (as Everett and Deutsch will do for
the many-world, but not the immortality question though.
I think that in the priority matter (a boring subject but then
friends said that I must defend myself a little bit more) the criteria
is the date of the publication. It is one thing to get an idea and a
different thing to publish it. You need to fçind the idea but also to

James Higgo found a 1986 publication by Euan Sqires that mentions the
immortality argument. Perhaps that's not too much earlier for you to
claim independent discovery in your 1988 paper. Still the point is,
its one of those ideas that's floating around anyway - in the ether,
so to speak.

Also the universal dovetailer idea is also one of those that is fairly
obvious, and might have been discovered a number of times independently.

In some ways, these ideas are too simple for the issue of priority to
be taken seriously. Perhaps, but the fame game is fickle
indeed. Famous people are often not famous for their most important
work. My most cited paper according to Google Scholar On complexity and
emergence doesn't contain any original ideas at all! (Its a digestion
of what I've read on the topics)

On the other hand your COMP ontological reversal idea is truly
unique. Hopefully you are right, and it goes down in history as your
greatest contribution to human knowledge.

Cheers

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### Re: Smullyan Shmullyan, give me a real example


On Sat, May 13, 2006 at 05:28:31PM +0200, Bruno Marchal wrote:

Le 12-mai-06, à 09:41, Kim Jones a écrit :

Bruno,

I almost understand this. Just expand a little

Kim

On 11/05/2006, at 9:00 PM, Bruno Marchal wrote:

Schmidhuber did leave the list by refusing explicitly the first-third
person distinction (which explain why his great programmer does not
need to dovetail).

I guess it is a delicate point, a key point though, which overlaps the
ASSA/RSSA distinction (that is: the Absolute Self Sampling Assumption
versus the Relative Self Sampling Assumption).

If you identify a conscious first person history with a third person
describable computation, it can be argued that an explanation for
physics can be given by Bayesian sort of anthropic reasoning based on
some  universal  probability distribution like Hall Finney's
Kolmogorovian UDist. Note tat this approach relies also on Church
Thesis. Here somehow the TOE will be a winning little program. I agree
that this would hunt away the third person white rabbits.

I disagree. The UDist comes from looking at the measure induced on a
set of descriptions (or computations if your prefer, although the two
are not equivalent), given a reference Turing machine U. This appears
to be a 3rd person description, but it need not be so. As I have
pointed out (but suspect it hasn't really sunk in yet), U can be
taken to be the observer erself. When done this way, there is a 1st
person universal distribution, with a corresponding 1st person Occam
razor theorem. And this implies the absence of 1st person white rabbits.

Cheers

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### Re: Smullyan Shmullyan, give me a real example



Le 12-mai-06, à 09:41, Kim Jones a écrit :

Bruno,

I almost understand this. Just expand a little

Kim

On 11/05/2006, at 9:00 PM, Bruno Marchal wrote:

Schmidhuber did leave the list by refusing explicitly the first-third
person distinction (which explain why his great programmer does not
need to dovetail).

I guess it is a delicate point, a key point though, which overlaps the
ASSA/RSSA distinction (that is: the Absolute Self Sampling Assumption
versus the Relative Self Sampling Assumption).

If you identify a conscious first person history with a third person
describable computation, it can be argued that an explanation for
physics can be given by Bayesian sort of anthropic reasoning based on
some  universal  probability distribution like Hall Finney's
Kolmogorovian UDist. Note tat this approach relies also on Church
Thesis. Here somehow the TOE will be a winning little program. I agree
that this would hunt away the third person white rabbits.

Despite the obvious appeal for such an approach, once we take into
account the fact that we cannot know in which computations we belong,
and that we are not aware of the delay of a universal dovetailer to
rich the computationally accessible computational states, then we
realize that we need to take into account the fact that almost all
programs which generate us are *big*. Our consciousness is somehow
distributed in the whole of the comp-platonia (a non comp structure!).
Here somehow the TOE could still be given by a little program, but it
needs a justification how it can win an infinite battle with the big
programs, and eliminate a vaster collection of first person white
rabbits. (BTW we are very close to Descartes fifth meditation if you
know. His malin génie generates first person hallucinations). All
this follows from the UDA (Universal Dovetailer Argument).

From empiry it could be that the winning little program describes some
quantum universal dovetailer, or an universal unitary transformation,
modular functor (topological quantum computer), etc. but all what I try
to explain is that such little program must be justified as being
invariant for some notion of first person (plural) observable taking
into account the infinities of infinite computations (once we make
explicit the comp (or weaker) assumption. By identifying first and
third person experience we need only one successful computation as an
explanation. By being aware of the 1-3 distinction we have to dovetail
on all computations and (re)defined reality as a relative measure on
the possible ways of glueing consistent first person experience; if
not, I'm afraid the mind body problem remains under the rug.

Hope that help a little bit. Don't hesitate to ask more explanations.
Just be patient if I don't answer so quickly.
Some more technical points will be made clearer through the deepening
of diagonalization, perhaps.

Critics from ASSA people are welcome!

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example


Bruno,

I almost understand this. Just expand a little

Kim

On 11/05/2006, at 9:00 PM, Bruno Marchal wrote:

Schmidhuber did leave the list by refusing explicitly the first-third
person distinction (which explain why his great programmer does not
need to dovetail).

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### Re: Smullyan Shmullyan, give me a real example



On Fri, 12 May 2006, Saibal Mitra wrote:

Einstein seems to have believed in ''immortal observer moments''.

In a BBC documentary about time it was mentioned that Einstein consoled a
friend whose son had died in a tragic accident by saying that relativity
suggests that the past and the future are as real as the present.

I'm sure Einstein would turn in his grave at your quoted expression. An
immortal moment is a contradiction in terms, unless it implies a second
time which passes as we contemplate first time embedded in 4-D
space-time.  Unfortunately a lot of popular discussion of space-time
implicitly invokes this spurious second time, because it is hard to
decouple the language of existence from the language of existence *in
time*. To believe, with Einstein, that all points in space-time are
equally real (because the relativity of simultaneity means that there is
no unique now) is quite the opposite of the nutty idea that all events
exist now --- not even wrong, from Einstein's point of view.

Einstein actually expressed this view in a letter of condolence to the
widow of his old friend Michele Besso. His words are worth quoting
accurately:

In quitting this strange world he has once again preceded me by just a
little. That doesn't mean anything. For we convinced physicists the
distinction between past, present, and future is only an illusion, however
persistent.

Later physicists, in particular John Bell, have pointed out that
relativity doesn't *prove* that now is an illusion, it just makes it
impossible to identify any objective now.

Not that any of this has anything to do with the sort of immortality
contemplated by Everett, which is not at all enticing: like the Sibyl in
classical myth, his immortality would not be accompanied by eternal
youth... a rather horrible fate.

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### Re: Smullyan Shmullyan, give me a real example


From: Patrick Leahy [EMAIL PROTECTED]
Sent: Friday, May 12, 2006 12:56 PM
Subject: Re: Smullyan Shmullyan, give me a real example

On Fri, 12 May 2006, Saibal Mitra wrote:

Einstein seems to have believed in ''immortal observer moments''.

In a BBC documentary about time it was mentioned that Einstein consoled
a
friend whose son had died in a tragic accident by saying that relativity
suggests that the past and the future are as real as the present.

I'm sure Einstein would turn in his grave at your quoted expression. An
immortal moment is a contradiction in terms, unless it implies a second
time which passes as we contemplate first time embedded in 4-D
space-time.  Unfortunately a lot of popular discussion of space-time
implicitly invokes this spurious second time, because it is hard to
decouple the language of existence from the language of existence *in
time*. To believe, with Einstein, that all points in space-time are
equally real (because the relativity of simultaneity means that there is
no unique now) is quite the opposite of the nutty idea that all events
exist now --- not even wrong, from Einstein's point of view.

Einstein actually expressed this view in a letter of condolence to the
widow of his old friend Michele Besso. His words are worth quoting
accurately:

In quitting this strange world he has once again preceded me by just a
little. That doesn't mean anything. For we convinced physicists the
distinction between past, present, and future is only an illusion, however
persistent.

Later physicists, in particular John Bell, have pointed out that
relativity doesn't *prove* that now is an illusion, it just makes it
impossible to identify any objective now.

Not that any of this has anything to do with the sort of immortality
contemplated by Everett, which is not at all enticing: like the Sibyl in
classical myth, his immortality would not be accompanied by eternal
youth... a rather horrible fate.

Thanks for the correction and the exact quote. I only vaguely remembered
what was said in the program. Of course, ''immortal observer moment'' is
indeed contradictory. The point is, of course, that ''now'' implies a
localization in time just like ''here'' implies localization in space. Just
like things that don't exist here but do exist elsewhere are ''real'' so
should things that don't exist now anymore but did exist at some time in the
past.

Saibal

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### Re: Smullyan Shmullyan, give me a real example



Le 11-mai-06, à 01:07, Russell Standish a écrit :

(Sadly, Everett's daughter Liz, in her later suicide note, said
she was going to a parallel universe to be with her father...)

Sadly, because this is based on a total misunderstanding of QTI, I
guess.

I guess and/or hope it was just a poetical way to express herself on
her suicide.

bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example



Hi George,

Bruno,

Thank you for still working on my post. I am working on the reply, in
particular designing the set of function or number that can be
diagonalized to generate a large number. I shall be busy this weekend
with family matters but I will reply to you in detail.

Take it easy we have all the time. In case of trouble don't hesitate to
ask for hints or supplementary explanations. This is said for everyone.

I agree that the idea of quantum suicide did not originate with
Tegmark,
even though he is the one who popularized it.

Sure. He wrote a beautiful paper. Note that he is unaware of the more
general and more simple (?) comp immortality.

The idea also came to me
independently in the early 1990's as I was pondering the Scroedinger
cat
experiment. What if I was the cat? How would I feel? What if I was the
scientist conducting the experiment and I was inside a larger box
enclosing the whole experiment? Would I feel the superposition? These
are very obvious questions to ask. This Scroedinger cat experiment
approximately dates to the 1920-1930's (?) and it is very well possible
that others have had the same thought.

I think Schroedinger used the cat for explaining a paradoxical feature
of QM, and I have not see suggestions by him that comp leads to either
many world or quantum immortality (as Everett and Deutsch will do for
the many-world, but not the immortality question though.
I think that in the priority matter (a boring subject but then
friends said that I must defend myself a little bit more) the criteria
is the date of the publication. It is one thing to get an idea and a
different thing to publish it. You need to fçind the idea but also to
get the nerves to make it public. I have not publish so much easy
readable (original) thing not to insist a little bit on this,
especially given that I still somehow paying a hard price for having
dare to work on such questions in the seventies. I suspect a little bit
Russell (notably in his book) to dismiss how much both the universal
dovetailer and comp-immortality was (and still is actually) original in
the TOE framework. Russell makes often (in posts and in his preprint
book) the confusion between the notion of Universal Machine,
Schmidhuber great programmer (which does not dovetail) and the
Universal Dovetailer. Those notions are related but are not at all
equivalent in the search for a TOE, neither extensionally, nor
intensionally (different programs and different functions).
I developed and defended those ideas very early. This explains in part
why I have been confronted with an obvious natural skepticism, and this
is why I have provided the logical analysis (well to be true I got this
one simultaneously as I explained in the 1988 paper: it is even the
reason why I have chose to do math and not physics). Actually the
heart of the matter explanations will consist in showing how much
universal dovetailing and Church thesis are non trivial notions.
complexity. You need training in diagonalization to doubt Church's
thesis!
The same with the comp first person indeterminacy. Probably my main
easy (for you in this quite open-minded list) discovery. Remember that
Schmidhuber did leave the list by refusing explicitly the first-third
person distinction (which explain why his great programmer does not
need to dovetail). It is not just a question of priority, it is a
question of getting the notions right before.
Another point. james Higgo told us explicitly that, despite quantum
suicide, Tegmark did not believe in the quantum immortality consequence
of the quantum hyp, showing the big nuance between the immortality and
suicide points, often confuse in posts or elsewhere. Oops I must leave
...

Best,

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example


Einstein seems to have believed in ''immortal observer moments''.

In a BBC documentary about time it was mentioned that Einstein consoled a
friend whose son had died in a tragic accident by saying that relativity
suggests that the past and the future are as real as the present.

Saibal

From: Russell Standish [EMAIL PROTECTED]
Sent: Thursday, May 11, 2006 01:07 AM
Subject: Re: Smullyan Shmullyan, give me a real example

On Wed, May 10, 2006 at 11:13:27PM +0100, Patrick Leahy wrote:

On who invented quantum suicide, the following is from the biography of
Hugh Everett by Eugene B. Shikhovtsev and Kenneth W. Ford, at
http://space.mit.edu/home/tegmark/everett/

Atheist or not, Everett firmly believed that his many-worlds theory
guaranteed him immortality: His consciousness, he argued, is bound at
each
branching to follow whatever path does not lead to death --- and so on
infinitum. (Sadly, Everett's daughter Liz, in her later suicide note,
said
she was going to a parallel universe to be with her father...)

Sadly, because this is based on a total misunderstanding of QTI, I guess.

The reference is to Everett's views in 1979-80, but there is no reason
to
suppose that Everett had only just thought of it at the time. On a
personal note, some time in the '80s I met one of Everett's co-workers
who
told me that Everett used to justify his very unhealthy lifestyle on
exactly these grounds. In our world, Everett died of a heart attack aged
52.

I have always assumed that John Bell was thinking along these lines when
he commented on Everett's theory:

But if such a theory was taken seriously it would hardly be possible to
take anything else seriously. (1981, reprinted in _Speakable
Unspeakable in Quantum Mechanics).

These dates all mesh with Don Page's anecdote about Ed Teller :
immortality consequences widely known, but rarely talked about by the
early '80s.

For that matter, this idea is implicit in Borges' story The Garden of
Forking Paths (written before 1941), which provides the epigraph to the
DeWitt  Graham anthology on The Many Worlds Interpretation.

==
Dr J. P. Leahy, University of Manchester,
Jodrell Bank Observatory, School of Physics  Astronomy,
Macclesfield, Cheshire SK11 9DL, UK
Tel - +44 1477 572636, Fax - +44 1477 571618

Very interesting. Its a shame my manuscript is already at the
printers, I would have loved this for my background info on QTI.

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### Re: Smullyan Shmullyan, give me a real example


On Tue, May 09, 2006 at 03:59:39PM +0200, Bruno Marchal wrote:

Schroedinger came up with his
cat's paradox. Tegmark came up with the quantum suicide experiment.

I came up first with the comp suicide, and much later after with the
quantum suicide and with the kill the user sort of quantum
computation, well before Tegmark, and I am not sure this has helped to
make my work more acceptable or comprehensible. For me quantum
suicide was a confirmation of the fact, easily derivable from comp,
that even for purely empirical reason we can doubt mortality, or
doubt that the mortality issue is simple, like so many materialist tend
to think.

James Higgo published a web page describing the history of quantum
suicide aka comp suicide. The notion obvious predates both Tegmark and
Marchal - and there is some anecdotal evidence that Edward Teller knew
about the argument in the early eighties. It appears to have been a
dirty little secret, which has only really been considered
acceptable talk in polite scientific circles in the last 10 years or so.

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Mathematics0425 253119 ()
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### Re: Smullyan Shmullyan, give me a real example


Le 10-mai-06, à 04:19, Russell Standish a écrit :

James Higgo published a web page describing the history of quantum
suicide aka comp suicide. The notion obvious predates both Tegmark and
Marchal - and there is some anecdotal evidence that Edward Teller knew
about the argument in the early eighties. It appears to have been a
dirty little secret, which has only really been considered
acceptable talk in polite scientific circles in the last 10 years or so.

I explain quantum suicide, and I use it to explain the comp immortality in:
Marchal B., Informatique théorique et philosophie de l’esprit. Actes du 3ème colloque international de l’ARC, Toulouse 1988.

I have presented orally the paper at Toulouse in 1987. The paper contains the movie graph argument, and a much earlier version of that paper contains the RE paradox, one of many version of the UDA. That earlier paper has been published in two parts later under the forms:

Marchal B., Mechanism and Personal Identity, proceedings of WOCFAI 91, M. De Glas  D. Gabbay (Eds), Angkor, Paris, 1991.
Marchal B., 1992, Amoeba, Planaria, and Dreaming Machines, in Bourgine  Varela (Eds), Artificial Life, towards a practice of autonomous systems, ECAL 91, MIT press.

Look, you can see my work as the given of a purely arithmetical (more generally lobian) reconstruction of Lucas-Penrose type of argument against mechanism. Only, such argument does not show that we are not machine but only that *in case* we are machine *then* we cannot know which machine we are, nor can we know which computational paths support us, and there is already an indeterminacy there. Then I illustrate that we  (I mean the (hopefully) lobian machines) can reflect that indeterminacy. You can see it as a generalization of Everett's embedding of the physicists in the physical world; where instead I embed the mathematician (actually some arithmetican) in the mathematical (arithmetical) world. In both case this makes sense only when we distinguished first person and third person discourse.

But now, my preceding point was just that the existence of the discourse about quantum suicide or quantum immortality, which appears from empirical reasons, confirms the general statement that comp implies that any machine looking at herself below its substitution level should discover empirically the indetermination about which computations which support her, from which the comp immortality follows.

Obviously (?) I am suspecting a big part of the physical emerges already from the impossible statistics on number relations once you mix addition and multiplication. The advantage of the self-referential approach  (just made easier by comp, but it works on many type of non-machine or generalized infinite machines) is that it provides at its roots a difference between the truth and the true discourses on those questions (got through G* \ G and its intensional variants), the arithmetical Hypostases as I am tempted to call them since I read Plotinus.

You can see what I am mainly trying to say as: oh look we can *already* interview a universal machine about fundamental questions. I illustrate this by interviewing a lobian machine on the logics of the communicable, knowable and bettable (by Universal Machines) pertaining  on verifiable propositions (here verifiable = accessible by the Universal Dovetailer.
The goal: extract the whole measure on the relative continuations. Not just the logic of certainty.
The problem at this stage is mathematical and concerns the existence of not of some Hopf algebra of trees capable of explaining how to renormalize in front of the arithmetical white rabbits.

Bruno

http://iridia.ulb.ac.be/~marchal/

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### Re: Smullyan Shmullyan, give me a real example


Bruno,

Thank you for still working on my post. I am working on the reply, in
particular designing the set of function or number that can be
diagonalized to generate a large number. I shall be busy this weekend
with family matters but I will reply to you in detail.

I agree that the idea of quantum suicide did not originate with Tegmark,
even though he is the one who popularized it. The idea also came to me
independently in the early 1990's as I was pondering the Scroedinger cat
experiment. What if I was the cat? How would I feel? What if I was the
scientist conducting the experiment and I was inside a larger box
enclosing the whole experiment? Would I feel the superposition? These
are very obvious questions to ask. This Scroedinger cat experiment
approximately dates to the 1920-1930's (?) and it is very well possible
that others have had the same thought.

George

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### Re: Smullyan Shmullyan, give me a real example



On who invented quantum suicide, the following is from the biography of
Hugh Everett by Eugene B. Shikhovtsev and Kenneth W. Ford, at
http://space.mit.edu/home/tegmark/everett/

Atheist or not, Everett firmly believed that his many-worlds theory
guaranteed him immortality: His consciousness, he argued, is bound at each
infinitum. (Sadly, Everett's daughter Liz, in her later suicide note, said
she was going to a parallel universe to be with her father...)

The reference is to Everett's views in 1979-80, but there is no reason to
suppose that Everett had only just thought of it at the time. On a
personal note, some time in the '80s I met one of Everett's co-workers who
told me that Everett used to justify his very unhealthy lifestyle on
exactly these grounds. In our world, Everett died of a heart attack aged
52.

I have always assumed that John Bell was thinking along these lines when
he commented on Everett's theory:

But if such a theory was taken seriously it would hardly be possible to
take anything else seriously. (1981, reprinted in _Speakable
Unspeakable in Quantum Mechanics).

For that matter, this idea is implicit in Borges' story The Garden of
Forking Paths (written before 1941), which provides the epigraph to the
DeWitt  Graham anthology on The Many Worlds Interpretation.

==
Dr J. P. Leahy, University of Manchester,
Jodrell Bank Observatory, School of Physics  Astronomy,
Macclesfield, Cheshire SK11 9DL, UK
Tel - +44 1477 572636, Fax - +44 1477 571618

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### Re: Smullyan Shmullyan, give me a real example


On Wed, May 10, 2006 at 11:13:27PM +0100, Patrick Leahy wrote:

On who invented quantum suicide, the following is from the biography of
Hugh Everett by Eugene B. Shikhovtsev and Kenneth W. Ford, at
http://space.mit.edu/home/tegmark/everett/

Atheist or not, Everett firmly believed that his many-worlds theory
guaranteed him immortality: His consciousness, he argued, is bound at each
infinitum. (Sadly, Everett's daughter Liz, in her later suicide note, said
she was going to a parallel universe to be with her father...)

Sadly, because this is based on a total misunderstanding of QTI, I guess.

The reference is to Everett's views in 1979-80, but there is no reason to
suppose that Everett had only just thought of it at the time. On a
personal note, some time in the '80s I met one of Everett's co-workers who
told me that Everett used to justify his very unhealthy lifestyle on
exactly these grounds. In our world, Everett died of a heart attack aged
52.

I have always assumed that John Bell was thinking along these lines when
he commented on Everett's theory:

But if such a theory was taken seriously it would hardly be possible to
take anything else seriously. (1981, reprinted in _Speakable
Unspeakable in Quantum Mechanics).

These dates all mesh with Don Page's anecdote about Ed Teller :
immortality consequences widely known, but rarely talked about by the
early '80s.

For that matter, this idea is implicit in Borges' story The Garden of
Forking Paths (written before 1941), which provides the epigraph to the
DeWitt  Graham anthology on The Many Worlds Interpretation.

==
Dr J. P. Leahy, University of Manchester,
Jodrell Bank Observatory, School of Physics  Astronomy,
Macclesfield, Cheshire SK11 9DL, UK
Tel - +44 1477 572636, Fax - +44 1477 571618

Very interesting. Its a shame my manuscript is already at the
printers, I would have loved this for my background info on QTI.

--

A/Prof Russell Standish  Phone 8308 3119 (mobile)
Mathematics0425 253119 ()
UNSW SYDNEY 2052 [EMAIL PROTECTED]
Australiahttp://parallel.hpc.unsw.edu.au/rks
International prefix  +612, Interstate prefix 02

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### Re: Smullyan Shmullyan, give me a real example


Tom wrote

My beside myself statement was a punny reference to self-reference.
I meant that I am looking forward to your post(s) with positive
eagerness.

Thanks. Also, I will follow your suggestion to force me writing little
post. I will first answer an older post by George (not to confuse with
Georges).

Le 25-mars-06, à 00:51, George Levy a écrit :

I have read or rather tried to read Smullyan's book. His examples are
totally fabricated. I will never meet the white knight in the island of
liars and truthtellers. I need examples which are relevant to life, at
least the way I understand it in the context of the many-worlds.

The role of the knight-knave island is just to give an easy way to
produce self-referential sentences? This is explain on page 48 in
Forever Undecided (hereafter FU).

Einstein (or maybe someone writing about relativity) came up with the
paradox of the travelling aging twin.

Was it not Langevin?

Schroedinger came up with his
cat's paradox. Tegmark came up with the quantum suicide experiment.

I came up first with the comp suicide, and much later after with the
quantum suicide and with the kill the user sort of quantum
computation, well before Tegmark, and I am not sure this has helped to
make my work more acceptable or comprehensible. For me quantum
suicide was a confirmation of the fact, easily derivable from comp,
that even for purely empirical reason we can doubt mortality, or
doubt that the mortality issue is simple, like so many materialist tend
to think.

Granted, I will never travel near the speed of light; I will never put
a
cat in a box equipped with a random and automatized killing device; and
I will not attempt suicide; my wife would just kill me. However, these
examples fired up my imagination: travelling near the speed of light,
existing in a superposition of state, surviving a nuclear bomb under

Smullyan's white knigth had the mission to teach me about the logic of
G
and G*. Sorry, he failed. The white knight does not fire up my

It is exactly with the diagonal principle (FU page 211) that the
logical role of the Knight-knave Island is eventually eliminated. Now
his diagonal principle arises in the context of his Godelized
Universe for which Smullyan don't provide motivation (it still look
like a fairy tale). What makes Universe *Godelized* is really Church
Thesis, and that is really the missing key in FU (and actually in the
whole work of Smullyan). I make those steps more transparent in my SANE
paper. have you print it? It should help.

However I do care about life, death and immortality. The many-world
does
seem to guarantee a form of immortality, at least according to some
interpretations.

Yes, we have discuss this a lot. I think most people agree on this in
the list, both with the quantum MWI, or with some all computations
exist. I think the most serious involved people in this list just
disagree on how to quantify the (quantum or comp) indeterminacy. Of
course progress have been made on the quantum part of that problem, but
hardly on the comp part, which is actually presupposed in the quantum
MWI (cf Everett, Deutsch, ...).

I consider this issue to be very relevant since sooner
or later each one of us will be facing the issue of death or of
non-death.

Be careful with such motivation because it could lead t wishful
thinking. I am not sure you can appreciate the comp lesson which
shows above all the abyssal googelplexity or our ignorance. But then
such an ignorance appears to have a mathematical shape capable of
providing information, but this leads today just to hard mathematical
questions.

I would like someone to come up with an extreme adventure story like
the
travelling twin, Schroedinger's cat, or Tegmark's suicide experiment to
illustrate G and G*.

I think the whole UDA (Universal Dovetailer Argument) and the movie
graph paradox (or argument), and/or Maudlin's Olympia device are going
in that direction. G and G* are just the tools for making this
technical enough so that it leads to testable propositions. UDA is
really the intuitive (and rigorous although informal) path for an
understanding of the reversal between physics and numbers, including
the showing of how hard the comp (im)mortality puzzle is.

For example this story would describe a close brush
with death.. It would create a paradox by juxtaposing 1) classical or
common sense logic assuming a single world,

It seems to me that the UDA just does that. Do you see that classical
physics is a priori untenable with comp once we take the 1-3 person
distinction into account? I mean Classical physics is just
epistemologically incompatible with common sense, once comp is assumed.

2) classical or common sense
logic assuming the many-world, and

See just above.

3) G/G* logic assuming the many-world.



### Re: Smullyan Shmullyan, give me a real example


Le 27-mars-06, à 06:09, George Levy a écrit :

I am looking forward to being diagonalized. I hope it won't hurt too much.

Asap. Meanwhile you could already medidate on my first diagonalization post here.
You can ask (out or online) any question including about notations or definitions:

http://www.mail-archive.com/everything-list@eskimo.com/msg01561.html

If you find that unreadable, tell me and I will think about other ways to present it, or links ...

Also: did you grasp in FU the notions of:

reasoner of type 1
reasoner of type 1*
reasoner of type 2
reasoner of type 3
reasoner of type 4

and

reasoner of type G ?

Bruno

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### Re: Smullyan Shmullyan, give me a real example



Bruno Marchal wrote:
<>
Le 25-mars-06,  00:51, George Levy a crit :

Smullyan's white knigth had the mission to teach me about the logic of
G
and G*. Sorry, he failed.

All right, but this is just because he miss Church Thesis and Comp. His
purpose actually is just to introduce you to Godel and Lob theorems,
not to computer science. The heart of the matter is that mathematical
systems (machines, angels, whatever)  cannot escape the diagonalisation
lemma, and so life for them is like the life of those reasoners
travelling on fairy knight Knave island with curious self-referential
question.
With comp *we* cannot escape those diagonal propositions.

I am looking forward to examples involving people being
diagonalized...hmmm Hilbert did come up with a thought experiment with
an infinite number of people lodged in a hotel actually we want to
go further than that and assume an infinite number of selves in the
many-worldOnce upon many times (Ils etaient des fois...), there
were several princesses...they looked into self referential magic
mirrorsand they lived ever after.

I would like someone to come up with an extreme adventure story like
the
travelling twin, Schroedinger's cat, or Tegmark's suicide experiment to
illustrate G and G*. For example this story would describe a close
brush
with death.. It would create a paradox by juxtaposing 1) classical or
common sense logic assuming a single world,

I think you miss the diagonalization
notion. I will work on that.

I am looking forward to being diagonalized. I hope it won't hurt too
much.

I will give you "real examples", but don't
throw out FU to quickly. \

OK.

George

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### Re: Smullyan Shmullyan, give me a real example



Le 25-mars-06, à 00:51, George Levy a écrit :

Dear members of the list, Bruno and those who understand G.

I have read or rather tried to read Smullyan's book. His examples are
totally fabricated. I will never meet the white knight in the island of
liars and truthtellers.

Nor will any Lobian machines. The knight/knave Island is just a trick
for having simple self-referential statements. I think you miss the
heart of the matter section and the godelized universe. Not your
fault and despite my love of Smullyan I am quasi willing to say
Smullyan miss it too. The missing piece is Church thesis. And then with
comp we can understand that lobian machines live in a Godelized
Universe. I will come back on this.

I need examples which are relevant to life, at
least the way I understand it in the context of the many-worlds.

OK, OK, I work on this since many years. Modal logics and Solovay's
theorems provides a tools for progressing, but this need some
understanding of computer science and mathematical logic.

Einstein (or maybe someone writing about relativity) came up with the
paradox of the travelling aging twin. Schroedinger came up with his
cat's paradox. Tegmark came up with the quantum suicide experiment.

Actually I came up before but this is anecdotical. But I have
elaborated it in the comp frame. It is the UDA. You have acknowledge
understanding it years ago. The interview of the lobian machine just
illustrate how we can already interview of universal machine on the UDA
question, and extract the logic of the physical propositions.

Granted, I will never travel near the speed of light; I will never put
a
cat in a box equipped with a random and automatized killing device; and
I will not attempt suicide; my wife would just kill me. However, these
examples fired up my imagination: travelling near the speed of light,
existing in a superposition of state, surviving a nuclear bomb under

Smullyan's white knigth had the mission to teach me about the logic of
G
and G*. Sorry, he failed.

All right, but this is just because he miss Church Thesis and Comp. His
purpose actually is just to introduce you to Godel and Lob theorems,
not to computer science. The heart of the matter is that mathematical
systems (machines, angels, whatever)  cannot escape the diagonalisation
lemma, and so life for them is like the life of those reasoners
travelling on fairy knight Knave island with curious self-referential
question.
With comp *we* cannot escape those diagonal propositions.

The white knight does not fire up my
However I do care about life, death and immortality. The many-world
does
seem to guarantee a form of immortality, at least according to some
interpretations. I consider this issue to be very relevant since sooner
or later each one of us will be facing the issue of death or of
non-death.

I thought you did understand that comp entails different forms of
immortality. The interveiw of the lobian machine makes it possible to
get more precise consequences, including testable one (some already
tested).

I would like someone to come up with an extreme adventure story like
the
travelling twin, Schroedinger's cat, or Tegmark's suicide experiment to
illustrate G and G*. For example this story would describe a close
brush
with death.. It would create a paradox by juxtaposing 1) classical or
common sense logic assuming a single world,

UDA shows rather directly the impossibility of single world.

2) classical or common sense
logic assuming the many-world,

?

and 3) G/G* logic assuming the many-world.

Assuming comp, the third person worlds are the computational histories.
An history is a computations as seen by from some internal point of
view. the fact that correct self-referential propositions obeys G and
G* makes it possible to describe those histories

What would the white knight do if he were living in the many-world?
What
kind of situations would highlight his talent to think in G. Would his
behavior appear to be paradoxical from our logical point of view?

The white knight, (well actually any Knight on the Knight Knave
Island!) are not even reasoners. None types of reasoner applies
including G.

The intuitive explanation why physics emerges from numbers and numbers'
dream is already given in the UDA. Smullyan just introduce the logics
of self-reference (the provable one, G, and the true one, G*). The
relation with our field is the content of one half of my posts (the
other half being UDA itself). I think you miss the diagonalization
notion. I will work on that. I will give you real examples, but don't
throw out FU to quickly. He makes something hard easy, but indeed
don't give to much motivations, except some allusions to AI here and
there.

Bruno

PS. I will answer other posts asap.

http://iridia.ulb.ac.be/~marchal/



### Smullyan Shmullyan, give me a real example


Dear members of the list, Bruno and those who understand G.

I have read or rather tried to read Smullyan's book. His examples are
totally fabricated. I will never meet the white knight in the island of
liars and truthtellers. I need examples which are relevant to life, at
least the way I understand it in the context of the many-worlds.

Einstein (or maybe someone writing about relativity) came up with the
paradox of the travelling aging twin. Schroedinger came up with his
cat's paradox. Tegmark came up with the quantum suicide experiment.
Granted, I will never travel near the speed of light; I will never put a
cat in a box equipped with a random and automatized killing device; and
I will not attempt suicide; my wife would just kill me. However, these
examples fired up my imagination: travelling near the speed of light,
existing in a superposition of state, surviving a nuclear bomb under

Smullyan's white knigth had the mission to teach me about the logic of G
and G*. Sorry, he failed. The white knight does not fire up my
However I do care about life, death and immortality. The many-world does
seem to guarantee a form of immortality, at least according to some
interpretations. I consider this issue to be very relevant since sooner
or later each one of us will be facing the issue of death or of non-death.

I would like someone to come up with an extreme adventure story like the
travelling twin, Schroedinger's cat, or Tegmark's suicide experiment to
illustrate G and G*. For example this story would describe a close brush
with death.. It would create a paradox by juxtaposing 1) classical or
common sense logic assuming a single world, 2) classical or common sense
logic assuming the many-world, and 3) G/G* logic assuming the many-world.

What would the white knight do if he were living in the many-world? What
kind of situations would highlight his talent to think in G. Would his
behavior appear to be paradoxical from our logical point of view?

George Levy

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