Consequences of ZFC+"$|x|\lt|y| \rightarrow |2^x|\lt|2^y|$"

Question

This question stems from https://mathoverflow.net/a/6594/22332 and really is summarized in the title: Has there been any research on the power of $|x|\lt|y| \rightarrow |2^x|\lt|2^y|$? It seems like it would be an obvious candidate for extending ZFC by "intuitive" axioms, so I'd assume there was research and it was seen to not hold much power, but even then I'd be curious about those results!

Edit to add to save some link-clicking: The cited answer states that the statement is independent of ZFC (and is, obviously, confirmed by GCH) and its negation follows from "Martin's Axiom, when CH fails."

It is the negation of the axiom that follows from MA, when CH fails, since MA+$\neg$CH implies $2^\omega=2^{\omega_1}$, which is a counterexample of the implication you mention. — Joel David Hamkins, Oct 12 '14 at 02:10
Many related links: one, two, three, and I'm guessing there are a few more. — Asaf Karagila, Oct 12 '14 at 03:10

score 6 · Answer 1 · answered Oct 12 '14 at 04:18

It follows from your axiom the following, which is known as ``Weak GCH''

Weak GCH (WGCH): For all infinite cardinals, $2^\lambda < 2^{\lambda^+}$.

In fact they are equivalent, and the $WGCH$ also implies your axiom. We have the following:

Theorem. $WGCH$ holds at $\lambda$ iff weak-$\Diamond$ on $\lambda^+$ holds.

For the undefined notions and many applications of $WGCH$ see the following Why the weak GCH is true! by John Baldwin.

Consequences of ZFC+"$|x|\lt|y| \rightarrow |2^x|\lt|2^y|$"

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