- Contemporary messages sorted: [ by date ] [ by thread ] [ by subject ] [ by author ] [ by messages with attachments ]

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

Date: Fri, 29 Nov 2002 18:58:38 -0800

Stephen Paul King writes:

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.

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

*
This archive was generated by hypermail 2.3.0
: Fri Feb 16 2018 - 13:20:07 PST
*