Le 23-sept.-06, ˆ 07:01, Russell Standish a Žcrit :


> Anything provable by a finite set of axioms is necessarily a finite 
> string of
> symbols, and can be found as a subset of my Nothing.


You told us that your Nothing contains all strings. So it contains all 
formula as "theorems". But a theory which contains all formulas as 
theorems is inconsistent.
I am afraid you confuse some object level (the strings) and 
theory-level (the theorems about the strings).

Perhaps the exchange is unfair because I react as a "professional 
logician", and you try to convey something informally. But I think that 
at some point, in our difficult subject, we need to be entirely clear 
on what we assume or not especially if you are using formal objects, 
like strings.



> I should note that the PROJECTION postulate is implicit in your UDA
> when you come to speak of the 1-3 distinction. I don't think it can be
> derived explicitly from the three "legs" of COMP.


I'm afraid your are confusing the UDA, which is an informal (but 
rigorous) argument showing that IF I am "digitalisable" machine, then 
physics  or the "laws of Nature" emerge and are derivable from number 
theory, and the translation of UDA in arithmetic, alias the interview 
of a universal chatty machine. The UDA is a "reductio ad absurdo".  It 
assumes explicitly consciousness (or folk psychology or grandma 
psychology as I use those terms in the SANE paper) and a primitive 
physical universe. With this, the 1-3 distinction follows from the fact 
that if am copied at the correct level, the two copies cannot know the 
existence of each other and their personal discourse will 
differentiate. This is an "illusion" of projection like the wave packet 
*reduction* is an "illusion" in Everett theory. The UDA reasoning is 
simple and the conclusion is that there is no primitive physical 
universe or comp is false. Physics emerges then intuitively from just 
"immaterial dreams" with subtle overlappings. The UDA does not need to 
be formalized to become rigorous. But having that UDA-result, we have a 
thoroughly precise way to extract physics (and all the other 
hypostases) from the universal interview. For *this* we need to be 
entirely specific and formal. That is why in *all* my papers (on this 
subject) I never separate UDA from the lobian interview. This is hard: 
I would not have succeed without Godel, Lob and other incompleteness 
theorems.
I have a problem with your way of talking because you are mixing 
informal talk with formal object (like the strings). Like when you 
write:


> The Nothing itself does not have any properties in itself to speak
> of. Rather it is the PROJECTION postulate that means we can treat it
> as the set of all strings, from which any conscious viewpoint must
> correspond to a subset of strings.


It looks like a mixing of UDA and the lobian UDA. It is too much fuzzy 
for me.


>
>> But it is neither "nothing". It is the natural numbers without 
>> addition
>> and multiplication, the countable order, + non standard models.
>
> I disagree - it is more like the real numbers without order, addition
> and multiplication group structures, but perhaps with the standard
> topology, since I want to derive a measure.


Are you saying that your Nothing is the topological line? Again it is 
not nothing (or it is very confusing to call it nothing), and what you 
intend will depend on your axiomatization of it. If you stay in first 
order logic, this will give an even weaker theory than the theory of 
finite strings: you will no more be able to prove the existence of any 
integer, or if you take a second order logic presentation of it, then 
your "nothing" will contain much more than what the ontic comp toes 
needs, and this is still much more than "nothing". To be franc I am 
astonished you want already infinite objects at the ontological level. 
If *all* infinite strings are in the ontology, that could be a 
departure from comp (and that would be interesting because, by UDA, 
that would make your theory predicting a different physics and then we 
could test it (at least in principle), and only when your theory will 
be precise enough.


> I don't know what Q1, Q2 and Q3
> are.


Robinson Arithmetic is formalized by the following set of axioms 
(written in first order language and in "french"):

Q1)   Ax        0 ­ s(x)    [0 is not a successor]
Q2)   AxAy    x ­ y -> s(x) ­ s(y)  [different numbers have different 
successors]
Q3)   Ax        (x ­ 0  ->  Ey(x = s(y))    [all numbers are successor, 
except 0]

Together with the definition of addition:

Q4)   Ax        x + 0  =  x [adding 0 to a number doesn't change it]
Q5)   AxAy   x + s(y)  =  s(x + y)  {adding some number x with a 
successor of some number y gives the successor of the addition of x and 
y]

and the definition of multiplication:

Q6)   Ax         x * 0  =  0 [multiplying a number by 0 gives 0]
Q7)   AxAy    x * s(y) = (x * y) + x  [if someone asks I will put this 
one in english but it is long and less understandable!]


PA, Peano Arithmetic, the "simplest" lobian machine for our purpose: is 
RA + the inductions axioms:
[ if P(0) and Ax (P(x) -> P(s(x)) ]  -> AxP(x)


RA is too dumb to be interviewed, but sufficiently clever to simulate 
the Universal Dovetailer, including the infinitely simulation of PA, 
and I interview those PAs. Unlike RA, PA has strong introspective 
power. PA can prove its own Godel and Lob theorems.
For example, if we let RA dovetail on all the proofs RA can build, then 
soon or later RA will prove that PA can prove the consistency of RA, 
but RA itself will never prove its own consistency. To infer that RA 
can prove its own consistency from the fact that RA proves that PA 
proves the consistency of RA would be like inferring I have a headache 
when I simulate Einstein's brain and got the sentence "I have a 
headache" (that is mainly Searle's error in his chinese room argument).

The distinction between RA and PA is introduced because we have to 
distinguish simulation and belief/theory. Why do we  *have to*? Because 
of incompleteness which is the roots of all those nuances in computer 
science, and from which I derive all hypostases or x-person point of 
view, including the one which gives the "comp-physics".


TO SUM UP:  the UDA is an informal rigorous, hopefully correct, 
argument showing that if we are machine then we are immaterial machine 
and that immateriality is contagious so that eventually (logically) 
physics is derivable from number theory under the form of an hypostase 
(the first person plural view).

The lobian interview *is* the extraction of physics (and of the other 
hypostases). It presupposes a complete understanding of the UDA 
consequence, and it extracts the comp-physics from assumptions on 
numbers only.

I though your ontic TOE (the strings) was similar to RA, but I guess I 
was wrong, so I am less sure I understand what you try to do.

Hope this helps,

Bruno


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


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