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From: Juergen Schmidhuber <juergen.domain.name.hidden>

Date: Wed, 31 Oct 2001 10:49:41 +0100

Cannibalizing previous thread "Provable vs Computable:"

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.

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?

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.

Such a virtual reality or universe is perfectly well-defined. We can

program it. At some point each history prefix will remain stable

forever.

Even if we know p and q, however, in general we will never know for sure

whether some q(n) that is still zero won't flip to 1 at some point,

because of Goedel etc. So this universe features lots of unprovable

aspects.

But why should this lack of provability matter? Ignoring this universe

just implies loss of generality. Provability is not the issue.

Juergen Schmidhuber

http://www.idsia.ch/~juergen/

http://www.idsia.ch/~juergen/everything/html.html

http://www.idsia.ch/~juergen/toesv2/

Received on Wed Oct 31 2001 - 01:50:41 PST

Date: Wed, 31 Oct 2001 10:49:41 +0100

Cannibalizing previous thread "Provable vs Computable:"

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.

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?

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.

Such a virtual reality or universe is perfectly well-defined. We can

program it. At some point each history prefix will remain stable

forever.

Even if we know p and q, however, in general we will never know for sure

whether some q(n) that is still zero won't flip to 1 at some point,

because of Goedel etc. So this universe features lots of unprovable

aspects.

But why should this lack of provability matter? Ignoring this universe

just implies loss of generality. Provability is not the issue.

Juergen Schmidhuber

http://www.idsia.ch/~juergen/

http://www.idsia.ch/~juergen/everything/html.html

http://www.idsia.ch/~juergen/toesv2/

Received on Wed Oct 31 2001 - 01:50:41 PST

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