Dear Bruno:
Thank you for your patience and the excellent response.
You should try to make your model part of established mathematics.
Not for the glory, but for making it comprehensible.
That is what I am trying to do here, but since I have proven to have too
few current mathematical skills
Dear Bruno:
I appreciate the conversation so I will try to build a common reference so
each additional step to my model can be built on that base and individually
commented on. As requested these are definitions and terms relevant to my
model not necessarily to established mathematics.
1a)
Dear Bruno:
At , you wrote:
Hal Ruhl wrote:
The assumption leads to a contradiction when String N exceeds the
complexity allowed by Chaitin. More information must be added to the
cascade for it to continue.
Why ? Only if your FAS produces as output just the string N
and then stop, then
Dear Juergen and Bruno:
Clearly I have a problem when I try to use mathematical terminology in
which I am not formally trained to explain my approach.
So here is an attempt to explain it in just a few normal words. My
system be it a FAS or not is modeled on the logistics equation process
Hal Ruhl wrote:
In what sense 4+1= is a proof chain ? A proof must be a sequence of
formula each of which are either axiom instance or theorems.
IMO it is ...
Definitions are not matter of opinion, but of conventional consensus.
... a sequence of:
1) A formula which in this case is of
Hi Hal,
Unfortunately I still miss some of your posts because of the absence of a
date stamp, my mailer puts them way at the top of my list since I save all
of some categories of e-mail.
I'm really sorry. I *will* come back to Eudora one day
Now I consider myself in favor of the idea
Hi Hal,
These are some of the things I want to explore as a result of formulating
my question re the UD.
To start:
1) Are you saying that the UD contains all other computations as data?
No. The UD is a program without data. It generates and executes all
computations. It dovetails, i.e. it
Dear Bruno:
Sorry I missed this. Here is my response.
At , you wrote:
Hal Ruhl wrote:
Juergen: Hal, here is an infinite chain of provable unique theorems:
1+1=2, 2+1=3, 3+1=4, 4+1=5, ...
First these are not theorems they are proof chains ending in theorems.
If you reinterpret
Hal Ruhl wrote:
1) The UD proof of the object all theorems is complex because each step
is a unique slice of progress towards some sub component of the target
object thus all steps are different and there are a great many of them.
2) The UD knows its proof is complex and since it is the only
Dear Russell:
I think you miss what I am saying.
At 4/20/01, you wrote:
I disagree. The UD will have a particular way of generating (or
enumerating) the theorems of the FAS, such that it doesn't generate
the same theorem twice.
The UD is [so it is said] generating all theorems. Some of these
But if P(A)=B and P'(B)=C are elegant proofs, it is very unlikely for
P'(P(A))=C to be an elegant proof. This is what the dovetailer is
constructing - it is not possible to know whether the any particular
proof output by the UD is elegant, only that it must contain elegant
proofs since it is
More detail:
Start with an axiom Aj, use it as data for a LISP expression Rj. A program
that computes the value of this expression B in AIT has length:
Expression + data + self-delimiter or in this case Rj + Aj +
Self-delimiter. Like this:
P = {Expression(data) + self-delimiter} computes
Here is a revised version of my comments on this subject. I think it fixes
several aspects of what I have had to say earlier.
Standalone deterministic evolving universes:
Such a universe is describable as a concatenation of single output programs
of the form:
Rj(Aj) - B; Rj(B) -
Dear Hal
Here is the second quote. It is from Chaitin's The Limits of Mathematics
page 90.
The first of these theorems states that an N-bit formal axiomatic system
cannot enable one to exhibit any specific object with a program-size
complexity greater than N + c.
Hal
Hal writes:
Here is a direct quote from page 24 of Chaitin's The Unknowable:
The general flavor of my work is like this. You compare the complexity of
the axioms to the complexity of the result you're trying to derive, and if
the result is more complex than the axioms, then you can not get
Hal Ruhl wrote:
Juergen: Hal, here is an infinite chain of provable unique theorems:
1+1=2, 2+1=3, 3+1=4, 4+1=5, ...
First these are not theorems they are proof chains ending in theorems.
If you reinterpret Juergen's word then you can tell him anything.
In all presentations of
Dear Hal
Since I was previously convinced by another that side bar discussions
should be avoided I will respond to this on the list.
At 4/12/01, you wrote:
Hal writes:
You are writing programs and they have a complexity. Chaitin limits this
complexity to no more than the complexity of the
Hal writes:
Well any assertion [object] with a LISP elegant program size greater than N
+ 356 can not be fully described by A since you can not identify its
elegant program with A.
Agreed.
Now Chaitin says on page 24 that he can not exhibit specific true,
unprovable assertions.
But
Dear Juergen:
In case what I tried to say was not clear the idea is that there are no
more than 2^(N + c) shortest possible unique proofs in an N-bit FAS. How
can number theory if it is a finite FAS contain an infinite number of
unique theorems?
Hal
Dear Jacques:
At 4/12/01, you wrote:
Maybe Hal, Russel and Jurgen should take this discussion to email and
just let us know how it turns out, because I get enough junk mail already.
I have run into those who do not like the side bar approach. I tend to
agree that it cuts all the others
Dear Juergen:
You demonstrate my point.
At 4/12/01, you wrote:
Hal, here is an infinite chain of provable unique theorems:
1+1=2, 2+1=3, 3+1=4, 4+1=5, ...
First these are not theorems they are proof chains ending in theorems.
For example:
4 + 1 = is a proof chain and the theorem proved is: 5
Dear Russell:
Yes we did indeed have a similar debate some time ago.
At that time I was still trying to express this point of view correctly and
admittedly made a number of mistakes back then [and still do].
Our debate helped me considerably and I thank you.
In response:
I just posted a
Dear Russell:
You wrote:
Why bound the proof?
It was not my idea. Chaitin equated complexity with a computing program's
length and a proof chain is a computing program according to Turing.
[rearranging your post]
1+1=2, 2+1=3, 3+1=4 ...
are all distinct theorems.
My view:
Again as in my
Dear Russell:
At 4/13/01, you wrote:
Bounded complexity does not imply bounded length. Examples include an
infinite sting of '0's, and the string '1234...9101112...'
That was part of the old debate and one of my initial mistakes. I am not
now talking about the length of theorems but the
Hal Ruhl wrote:
Dear Russell:
At 4/13/01, you wrote:
Bounded complexity does not imply bounded length. Examples include an
infinite sting of '0's, and the string '1234...9101112...'
That was part of the old debate and one of my initial mistakes. I am not
now talking about the length
Dear Juergen:
At 4/11/01, you wrote:
Hal, Chaitin just says you cannot prove 20 pound theorems with 10 pound
axioms.
Please refer to Chaitin's The Unknowable generally and page 25, Chapter
V, and note 10 at the bottom of page 97 in particular.
But the infinite cascade of all provable theorems
Hal, you wrote:
I believe that attempting an extensive detailed formal description of
the Everything is the wrong approach. IMO - at least in this case - the
more information used to describe, the smaller the thing described.
I was not able to follow this, but informal and vague descriptions
I appreciate Juergen's view. In essence he is assuming a nonuniform
distribution on the ensemble of descriptions, as though the
ensemble of descriptions are produced by the FAST algorithm. This is
perhaps the same as assuming a concrete universe.
In my approach (which didn't have this technical
Where does all the randomness come from?
Many physicists would be content with a statistical theory of everything
(TOE) based on simple probabilistic physical laws allowing for stochastic
predictions such as We do not know where this particular electron will
be in the next nanosecond, but with
Dear Juergen:
In reply:
Where does all the randomness come from?
Many physicists would be content with a statistical theory of everything
(TOE) based on simple probabilistic physical laws allowing for stochastic
predictions such as We do not know where this particular electron will
be in the
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