Re: "Everything" need a little more than 0 information

From: Hal Finney <hal.domain.name.hidden>
Date: Fri, 29 Nov 2002 18:58:38 -0800

Stephen Paul King writes:
> The set of all descriptions has at least the cardinality of the Reals by
> the Diagonalization argument by definition. Please recall how Cantor used
> the Diagonalization argument to prove that the Reals had a "larger"
> cardinality that that of the integers. If the Set of all Descriptions is all
> inclusive then it must containt any description that is constructable using
> "pieces" of each and every other description and thus can not has the same
> cardinality as that of the integers.

That would be true IF you include descriptions that are infinitely long.
Then the set of all descriptions would be of cardinality c. If your
definition of a description implies that each one must be finite, then the
set of all of them would have cardinality aleph-zero.

What Russell wrote was that the set of all descriptions could be computed
in c time on an ordinary Universal Turing Machine. My question is, does
it make sense to speak of a machine computing for c steps; it seems like
asking for the "c"th integer.

> I have a question: Where does Cantor's continuum hypothesis apply to
> this? (if at all)

This is the hypothesis that there is no transfinite cardinal between
aleph-zero and c, and I don't see any relevance to it.

Hal
Received on Fri Nov 29 2002 - 21:59:31 PST