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 [EMAIL PROTECTED] wrote:

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

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