Re: Does provability matter?

From: Wei Dai <weidai.domain.name.hidden>
Date: Fri, 2 Nov 2001 16:19:24 -0800

On Wed, Oct 31, 2001 at 10:49:41AM +0100, Juergen Schmidhuber wrote:
> Which are the logically possible universes? Tegmark mentioned a
> somewhat
> vaguely defined set of ``self-consistent mathematical structures,''
> implying provability of some sort. The postings of Marchal also focus
> on what's provable and what's not.

Do you understand Marchal's postings? I really have trouble making sense
of them. If you do, would you mind rephrasing his main ideas for me?

> Is provability really relevant? Philosophers and physicists find it
> sexy for its Goedelian limits. But what does this have to do with the
> set of possible universes?

What does it mean for a universe to be "provable"? One interpretation is
that there exists an algorithm to compute any finite time-space region of
the universe that always halts. Is that what you mean?

> The provability discussion seems to distract from the real issue. If we
> limit ourselves to universes corresponding to traditionally provable
> theorems then we will miss out on many formally and constructively
> describable universes that are computable in the limit yet in a certain
> sense soaked with unprovability.
>
> Example: a never ending universe history h is computed by a finite
> nonhalting program p. To simulate randomness and noise etc, p invokes
> a short pseudorandom generator subroutine q which also never halts. The
> n-th pseudorandom event of history h is based on q's n-th output bit
> q(n)
> which is initialized by 0 and set to 1 as soon as the n-th statement in
> an ordered list of all possible statements of a formal axiomatic system
> is proven by a theorem prover that systematically computes all provable
> theorems. Whenever q modifies some q(n) that was already used in the
> previous computation of h, p appropriately recomputes h since the n-th
> pseudorandom event.

Living in this universe is like living in a universe with a
halting-problem oracle, right? If you want to know whether program x
halts, you just measure the n-th pseudorandom event, where n is such that
the n-th theorem is "x halts." Yes?

The problem with universes with halting-problem oracles (which I'll just
call oracle universes) is that either the oracle is not exploitable
(for example if no one in the universe figures out that the pseudorandom
events actually correspond to theorems) in which case the universe just
looks like an ordinary "provable" universe, or the oracle is
exploitable, in which case the outcome becomes completely unimaginable for
us. (Or is that not true? Can we imagine what would happen if an
intelligent species came across an exploitable halting-problem oracle?)

> But why should this lack of provability matter? Ignoring this universe
> just implies loss of generality. Provability is not the issue.

If oracle universes could exist, how significant are they? What's
the probability that we live in such a universe, and how could we try to
determine and/or exploit this fact?
Received on Fri Nov 02 2001 - 16:21:19 PST