Within ZFC, is $2^{\aleph_0}<2^{\aleph_1}$ provable/independent?

Question

So, I ask whether from the ZFC axioms one can prove X that every uncountable set has strictly more than continuum many subsets, or whether X is independent of the ZFC axioms. Note that (within ZFC) the continuum hypothesis implies X and hence "not X" is not provable in ZFC if ZFC is consistent.

The above question arose from this MO-question to which an answer is given by Problem 10 on page 99 in Richard M. Dudley's book Real Analysis and Probability, Wadsworth 1989. However, in view of Problem 9 on page 387 loc.cit. it seems that some additional hypothesis (e.g. CH or the weaker X above) is missing in Dudley's problem setting. If X is provable within ZFC, then without any additional assumptions we have the result that every Borel map from any separable metric space to a metric space has separable range. If X is not provable, then Dudley's hints for a possible proof seem to require some ad hoc assumption for the proof to succeed.

Answer 1

The assertion that $2^{\aleph_0}=2^{\aleph_1}$ is known as Luzin's hypothesis, and was presented by Luzin as an alternative to Cantor's continuum hypothesis.

This is now known to be independent of ZFC by the method of forcing (assuming ZFC is consistent).

Namely, under the GCH, which is true in the constructible universe, we have $2^{\aleph_0}=\aleph_1<2^{\aleph_1}$ and therefore the negation of Luzin's hypothesis.

But in Cohen's model of $\neg\text{CH}$, obtained by forcing over $L$ to add $\aleph_2$ many Cohen reals, Luzin's hypothesis is true. The basic reason is that one can count in the ground model the number of "nice" names for subsets of $\aleph_1$ in the forcing extension, and there are only $\aleph_2$ many such names in the ground model. So $2^{\aleph_1}=\aleph_2$ in the Cohen model.

One can produce in just the same way models of ZFC in which all the $2^{\aleph_n}$'s are equal for finite $n$, or indeed, where $2^\kappa=2^{\aleph_0}$ for all cardinals $\kappa<2^{\aleph_0}$. Indeed, this situation is a consequence of Martin's axiom, which can hold even when $2^{\aleph_0}$ is quite large, as large as desired.

Models where Luzin's hypothesis is true provide counterexamples to the powerset size axiom, the principle asserting $$\kappa<\lambda\implies2^\kappa<2^\lambda$$ This is a principle that many people find very natural, but it is independent of ZFC.

But Luzin's hypothesis is not equivalent to the failure that principle, since by Easton's theorem, we can by forcing control the continuum function $\kappa\mapsto 2^\kappa$ on the infinite regular cardinals in a very flexible manner.

For example, we could have the GCH holding up to $\aleph_{17}$, but then $2^{\aleph_n}=\aleph_{\omega_1+5}$ for all natural numbers $n>17$.