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

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Herman Rubin

Posts: 399
Registered: 2/4/10
Re: question on set theory
Posted: Jan 2, 2013 2:15 PM
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On 2013-01-02,
<> 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?

Yes, and rather easily. With C, we have the trichotomy,
either X < Y, X ~ Y, or X > Y. If not X < Y, P(X) >= P(Y).

> Thanks,
> Mike

This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University Phone: (765)494-6054 FAX: (765)494-0558

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