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From: <juergen.domain.name.hidden>

Date: Thu, 21 Jun 2001 11:24:25 +0200

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/

Received on Thu Jun 21 2001 - 02:26:39 PDT

Date: Thu, 21 Jun 2001 11:24:25 +0200

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/

Received on Thu Jun 21 2001 - 02:26:39 PDT

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