Does $2^c=2^d \implies c=d$ for all cardinals $c$ and $d$ imply GCH?

Question

Suppose we add the statement "$2^c=2^d \implies c=d$, for all cardinals $c$ and $d$" to the ZFC axioms. Would we then be able to derive the GCH, the generalized continuum hypothesis?

Answer 1

The answer is no. It is consistent with $\mathsf{ZFC}$ that $2^{\aleph_n}=\aleph_{n+2}$ for all $n<\omega$ and GCH holds above $\aleph_\omega$, and you can derive the injectivity of the continuum function from this assumption. Its consistency follows from Easton's theorem, or you may force it directly from the model of $\mathsf{ZFC+GCH}$.

We can have even more from Easton's theorem: we can have $2^{\kappa}=\kappa^{++}$ for all regular cardinals $\kappa$ and $2^\kappa=\kappa^+$ for singular cardinals. You can also see that we have injectivity of the continuum function from this situation.

You may ask your assumption proves $\mathsf{GCH}$ at least for singular cardinals, and the answer is still negative if you assume large cardinal axioms. It is a result of Woodin that $2^\kappa=\kappa^{++}$ is consistent for all cardinals $\kappa$ if we have a supercompact cardinal (see page 2 of this paper).