Re: Is the universe computable?

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Mon, 12 Jan 2004 15:03:31 +0100

At 15:42 09/01/04 -0500, Jesse Mazer wrote:
>Bruno Marchal wrote:
>
>>I don't think the word "universe" is a basic term. It is a sort
>>or deity for atheist. All my work can be seen as an attempt to mak
>>it more palatable in the comp frame.
>>Tegmark, imo, goes in the right direction, but seems unaware
>>of the difficulties mathematicians discovered when just trying to
>>define the or even a "mathematical universe". Of course tremendous
>>progress has been made (in set theory, in category theory) giving
>>tools to provide some *approximation*, but the big mathematical
>>whole seems really inaccessible. With comp it can be shown
>>(first person) inaccessible, even unnameable ...
>
>"Inaccessible" in what sense? How do you use comp to show this? If this is
>something you've addressed in a previous post, feel free to just provide a
>link...


This is a consequence of Tarski theorem. Do you know it?
I think I have said this before but I don't find the links (I have to much
mentioned
the result by McKinsey and Tarski in Modal logic, so searching the archive
with "tarski" does not help).
Let me explain it briefly. With the platonist assumption being a part
of the comp hyp, we can identify in some way truth and reality (in a very
large sense which does not postulate that reality is necessarily
physical reality). That is "Reality" is identified with the set of all true
propositions
in some rich language.
Now Tarski theorem, like Godel's theorem, can be applied to any
"sufficiently rich" theory or to any sound machine. Tarski theorem says that
there is no truth predicate definable in the language of such
theories/machines.
Nor is "knowledge" definable for similar reason. So any complete platonist
notion of truth or knowledge cannot be defined in any language used by the
machine,
strictly speaking such vast notion of truth is just inaccessible by the
machine,
and this despite the fact a machine can build transfinite ladder of better
approximation
of it. By way of contrast the notion of "consistency" *is* definable in the
language of the
machine, only themachine cannot prove its own consistency (by Godel), but
the machine
can express it. Now, with Tarski the machine cannot even express it.

Like Godel's theorem, tarski theorem is a quasi direct consequence of the
*diagonalisation lemma":

For any formula A(x), there is a proposition k such that the machine
will prove "k <-> A(k)". Note: A(k) is put for the longer A(code of k)

In case a truth predicate V(x) could be defined in arithmetic or in the
machine's
tong, the machine would be able to define a falsity predicate (as -V(x) ), and
by the diagonalisation lemma, the machine would be able to prove the
"Epimenid sentence" "k <-> -V(k), which is absurd V being a truth predicate.

Truth, or any "complete description" of reality cannot have a definition,
or a name:
semantical notion like truth or knowledge are undefinable (unnameable).

Actually we don't really need comp in the sense that these "limitation theorem"
applies to much powerful theories or "divine" machine with oracle, ...

OK?

Bruno
Received on Mon Jan 12 2004 - 09:10:17 PST