The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math.research

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: question on set theory
Replies: 4   Last Post: Jan 4, 2013 2:17 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Dan Luecking

Posts: 26
Registered: 11/12/08
Re: question on set theory
Posted: Jan 2, 2013 3:30 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Wed, 2 Jan 2013 17:37:20 +0000 (GMT),
<> wrote:

>Happy New Year everybody!
> For a set X I will write X < Y to indicate that the cardinality of X is strictly smaller than the cardinality of Y. Let me write P(X) for the power set of X. Consider the proposition:
>Prop: If P(X) < P(Y)then X < Y.
>Is it possible to prove this in ZFC (without continuum hypothesis)? If not is it perhaps the case that the Prop is equivalent to generalized continuum hypothesis? It is trivial to show that generalized continuum hypothesis implies the proposition, but what about the other direction?

This should only require the axiom of choice:

Let X ~ Y mean X and Y are in 1-1 correspondence.
One defines < as follows:
First: X <= Y means X is in 1-1 correspondence with a subset of Y
Then: prove (I think AC is required):
if X <= Y and Y <= X then X ~ Y.
Then define X < Y to mean X <= Y but not X ~ Y.

This yields the tricotomy property: Either X < Y or Y < X or X ~ Y
(exclusive or).

Now if P(X) < P(Y) then there are three cases for X and Y
1. X ~ Y. In this case that 1-1 correspondence of X and Y
induces a 1-1 corespondence between subsets
(elements of P(X)) so P(X) ~ P(Y) and this case is out.
2. Y < X. This means Y is in 1-1 corespondence with a subset
of X. As in case 1, this provides a 1-1 correspondence of
P(Y) with a subset of P(X) so P(Y) < P(X) or P(Y) ~ P(X)
and this case is out.
3. X < Y is the only case remaining.

To reply by email, change LookInSig to luecking

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.