On 01/03/13 02:22, email@example.com wrote: > On Wednesday, January 2, 2013 3:30:01 PM UTC-5, Dan Luecking wrote: >> On Wed, 2 Jan 2013 17:37:20 +0000 (GMT), >> > > Apologies everyone. I posted the question late at night and wrote > the proposition backwards. I meant to say consider > > Prop: If X< Y then P(X)< P(Y) > > Can that be proved in ZFC? If not is it perhaps equivalent to > generalized continuum hypothesis? It is easy to show in ZF with or > without choice that that X< Y implies P(X)<=P(Y). But can it > possibly be the case that X< Y but P(X)=P(Y)? > > Mike > >
No, and no (unless of course you want to consider the possibility that ZFC is inconsistent, in which case anything would be provable in ZFC). Assuming ZF(C) is consistent, so are ZFC plus the failure of the continuum hypothesis plus either your proposition or its negation (where in the failure X can be taken to be the set of natural numbers); this follows from well-known (old) results of Easton about the behavior of the continuum function (which maps the cardinality of a set to the cardinality of its power set) at regular cardinals together with well-known (old) results of Jensen constraining its behavior at singular cardinals.
There are a few propositions about the continuum function easily provable in ZFC: (1) Cantor's theorem that for any set X, |X| < |P(X)| (where of course P denotes power set and |.| denotes cardinality), (2) for any sets X and Y, |X| <= |Y| implies |P(X)| <= |P(Y)|, and (3) for any set X, P(X) is not the union of |X| or fewer sets each of cardinality less than |P(X)|. Easton essentially showed (assuming ZFC is consistent) that nothing else can be proved in ZFC about the mapping from |X| to |P(X)| for regular cardinalities |X| (i.e. for the case in which X is not the union of fewer than |X| sets each of cardinality less than |X|). In particular, we can either have |P(N)| = |P(Y)| for some uncountable Y (one can take Y to be a subset of the reals, though by (1) Y must then have cardinality less than that of the reals), in which case your proposition and the continuum hypothesis both fail, or |P(N)| > |R| (so the continuum hypothesis fails) and yet your proposition holds.
Cardinal arithmetic is interesting in that it has really elementary results living cheek-by-jowl with incredibly deep theorems.