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I was very surprised to read on Wikipedia that it is undecidable in ZFC whether "$X\prec Y$ implies $\wp X\prec \wp Y$", see this link wiki. I tried to find out more but could not find any papers on this result. I would have thought this statement is as fundamental as CH or GCH and so should be well discussed.

Can anyone cast some light on this, if this statement were taken as an axiom what would be the consequences, is it related to GCH, and is there a good paper or article on it.

Mark Kortink
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Yes, this is a well-known undecidable statement. It is implied by GCH (which says that $2^\kappa=\kappa^+$, so if $\kappa<\lambda$ then $2^\kappa<2^\lambda$ since $\kappa^+<\lambda^+$). See Easton's Theorem, which - among other things - implies that it is consistent with ZFC that $2^{\aleph_0}=2^{\aleph_1}=\aleph_2$, as well as many similar variations. (Actually, this result holds in Cohen's original model of ZFC+$\neg$CH, although I don't know if this was known prior to Easton's result. EDIT: See Andreas' comment below.)

One reason you won't see much written about this statement is that - unlike CH or GCH - it doesn't seem to have a lot of useful consequences. It's just not very strong - in particular, note that it does not imply GCH. Indeed, we can have GCH fail everywhere and have this statement still hold; see https://mathoverflow.net/questions/138308/woodins-unpublished-proof-of-the-global-failure-of-gch, which describes a result that (assuming large cardinals) it is consistent that $2^\kappa=\kappa^{++}$ for every $\kappa$.

Perhaps surprisingly, the negation of this statement follows from a useful set-theoretic principle, namely the Proper Forcing Axiom which implies $2^{\aleph_0}=2^{\aleph_1}=\aleph_2$. The Proper Forcing Axiom, among other things, implies that there is a collection of five linear orders of size $\aleph_1$ such that any linear order of size $\aleph_1$ contains a suborder isomorphic to one of these five; that any two $\aleph_1$-dense subsets of $\mathbb{R}$ are isomorphic; and a number of other nice things.

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Noah Schweber
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  • I'm not sure whether Cohen knew that $2^{\aleph_0}=2^{\aleph_1}=\aleph_2$ in the model obtained from $L$ by adjoining $\aleph_2$ Cohen reals, but I'm quite confident that Solovay knew it before Easton began to work on this topic. (By the way, I wrote "from $L$" rather than "from a model of GCH", because, in Cohen's original work, forcing was ramified forcing over models of ZF + V=L.) – Andreas Blass Aug 22 '16 at 03:23
  • @AndreasBlass Ah yes - I'd forgotten about Solovay's work preceding Easton's theorem. My memory is that Solovay showed that $2^{\aleph_0}$ could be anything of uncountable cofinality, but did not look at $2^\kappa$ for uncountable $\kappa$, and that Easton was the first person to do so; is that right? – Noah Schweber Aug 22 '16 at 03:25
  • I don't remember whether Solovay's "anything it ought to be" paper said anything about $2^\kappa$ for uncountable $\kappa$, but in proving that one gets the desired result for $2^{\aleph_0}$ he must have developed the "counting names" argument, which is all you need to evaluate $2^{\aleph_1}$ in this model. – Andreas Blass Aug 22 '16 at 03:32
  • Thanks very much for the answer. I guess my point of view is that most "axioms" about cardinals like CH, GCH, large cardinal axioms, are tentative because our intuition has no basis at very large infinities. Perhaps CH is now tamed, surely we would have found a counter-example in $\mathbb R$ by now if not. But is seems to me "$X\prec Y$ implies $\wp X \prec \wp Y$" is a cardinal axiom that is intuitively obvious based on the primitive conception of what a set is. In that way it is similar to AC. Shame it's not fruitful. – Mark Kortink Aug 23 '16 at 00:27
  • @MarkKortink Re: your comment about CH: note that a version of that was already known long before Cohen: namely, that every uncountable Borel (indeed, analytic) set of reals has size continuum. Since Borel = "infinitarily first-order definable" (this can be made precise), this shows that no "nicely definable" set of reals can be a counterexample to CH; and this can be pushed into the projective hierarchy via large cardinals (but this was post-Cohen). Re: the intuitiveness of the principle you describe, I would suggest looking at PFA, which (to me) is very intutive but contradicts the above. – Noah Schweber Aug 23 '16 at 00:48