From: Perry E. Metzger <perry@piermont.com>

Date: Tue Dec 09 1997 - 19:53:44 PST

Date: Tue Dec 09 1997 - 19:53:44 PST

Hal Finney writes:

*> In the case of the computation, suppose every time you solve one problem
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*> you generate 10 new ones. You still are able to get to every problem
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*> eventually. You generated problems 0-9 at time 0, problems 10-19 at time
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*> 1, problems 20-29 at time 2, and in general problems 10n - 10n+9 at time n.
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*> You are still able to retire problem x at time x, so every problem gets
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*> dealt with in its turn and there are no problems which never get solved.
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You are making an implicit assumption, which is that the number of

problems is countably infinite. It is not. Cantorian set theory shows

that many sets are NOT mappable into the integers.

The computing cycles available in the universe, even if infinite, are

mappable into the integers. However, it is trivial to state classes of

problems that map into the reals, and a trivial diagonalization

argument shows you can't map the one into the other.

Perry

Received on Wed Dec 10 03:45:31 1997

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