Is cardinal exponentiation strictly monotone in the exponent?

Question

Let $\kappa$ be a cardinal, does $2^\kappa<2^{\kappa^+}$ always hold? It clearly holds if one assumes generalized continuum hypothesis, but does it also hold if one assumes otherwise?

Answer 1

This is independent of $\mathsf{ZFC}$. It consistently holds, because it holds under $\mathsf{GCH}$, and it consistently doesn't hold, because a two step iteration of Cohen forcing over a model of $\mathsf{GCH}$ can be used to force $2^\omega=2^{\omega_1}$.