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From: Brent Meeker <meekerdb.domain.name.hidden>

Date: Thu, 21 Jun 2001 08:40:52 -0700 (PDT)

Well put Juergen. The question arose as to whether our universe could be

or continous. Don't the computable numbers form a continuum; hence even

restricting the universe to one we can describe would still allow it to be

continuous?

Brent Meeker

On Thu, 21 Jun 2001 juergen.domain.name.hidden wrote:

*> There has been some confusion regarding the various kinds of infinity.
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*>
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*> There is the rather harmless kind: the countable one. And some say there
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*> is another kind, a strange one, the one associated with the uncountable
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*> continuum, the one whose very existence many deny.
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*>
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*> Do not lump them together.
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*>
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*> The former is accessible by nonhalting computer programs. The latter is not.
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*>
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*> A program that runs forever cannot consume more than countable time steps
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*> and storage cells, adding a new cell whenever it needs one. For example,
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*> countably many steps and cells are sufficient to print all digits of
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*> the real number Pi. Therefore Pi is "computable in the limit."
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*>
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*> But countable time and space resources are NOT sufficient to print all
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*> digits of all random reals in the continuum. In fact, countable resources
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*> are not even sufficient to print all (countably many) digits of a single
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*> real without finite individual description. Unlike Pi, such truly random
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*> reals (and almost all reals are truly random) are NOT computable in the
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*> limit, although all their finite beginnings are.
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*>
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*> Pi has a finite description. Are all infinite objects with finite
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*> descriptions computable in the limit? No. Counter example: Infinite Omega
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*> (the halting probability of a universal Turing machine with random input)
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*> has a finite description, but countable resources are NOT sufficient
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*> to print it, although each finite beginning of Omega is computable in
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*> the limit.
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*>
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*> Algorithmic TOEs are limited to the comparatively few, countably many,
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*> possibly infinite universe histories with finite descriptions. ATOEs deny
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*> the existence of other histories, of histories we cannot even describe
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*> in principle, of histories we cannot reasonably talk about.
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*>
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*> Likewise, ATOEs are restricted to probabilities computable in the limit.
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*> We cannot formally describe other probabilities, and therefore cannot
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*> write reasonable papers about them. This apparently harmless restriction
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*> is the very reason that any complex future (among all the possible
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*> futures compatible with the anthropic principle) necessarily is unlikely.
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*>
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*> Juergen Schmidhuber
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*>
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*> http://www.idsia.ch/~juergen/
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*> http://www.idsia.ch/~juergen/everything/html.html
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*> http://www.idsia.ch/~juergen/toesv2/
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*>
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*>
*

Received on Thu Jun 21 2001 - 08:41:44 PDT

Date: Thu, 21 Jun 2001 08:40:52 -0700 (PDT)

Well put Juergen. The question arose as to whether our universe could be

or continous. Don't the computable numbers form a continuum; hence even

restricting the universe to one we can describe would still allow it to be

continuous?

Brent Meeker

On Thu, 21 Jun 2001 juergen.domain.name.hidden wrote:

Received on Thu Jun 21 2001 - 08:41:44 PDT

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