On 2013-01-02, email@example.com <firstname.lastname@example.org> 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 email@example.com Phone: (765)494-6054 FAX: (765)494-0558