Easton's theorem

Remarks

The two conditions are exactly the constraints that ZFC proves for $\kappa \mapsto 2^\kappa$ on regular cardinals. Monotonicity is clear, and $\operatorname{cf}(2^\kappa) > \kappa$ is König's theorem. The proof uses a class product of Cohen-style forcings, one for each regular $\kappa$, with supports chosen so that the whole product preserves cardinals. Example: one can have $2^{\aleph_0} = \aleph_2$ and $2^{\aleph_1} = \aleph_2$, or $2^{\aleph_0} = \aleph_{\omega + 1}$. But $2^{\aleph_0} = \aleph_\omega$ is impossible, since $\operatorname{cf}(\aleph_\omega) = \omega$. The theorem says nothing about singular cardinals, where the situation is far more constrained, as in Silver's theorem and the singular cardinals hypothesis.

Going deeper

For a student

Forcing with Cohen reals shows that $2^{\aleph_0}$ can be any cardinal of uncountable cofinality. Easton's theorem extends this to all regular cardinals at once. It shows that monotonicity and König's theorem are the only ZFC constraints on the function $\kappa \mapsto 2^\kappa$ on regular cardinals. At singular cardinals the story changes: the values of $2^\kappa$ below a singular cardinal can restrict its value at that cardinal. This is the subject of Silver's theorem, of the singular cardinals hypothesis and of Shelah's pcf theory.