Necessary constraints on the regular-cardinal continuum function

Assume [F4]. Let $\kappa$ be an infinite cardinal. By [F1], $2^{\kappa }=\mid \mathcal{P}(\kappa )\mid$ is a cardinal and $\kappa <2^{\kappa }$. Restricting to infinite regular $\kappa$ gives clause (a).

Let $\kappa \le \lambda$ be infinite regular cardinals. Every subset of $\kappa$ is a subset of $\lambda$, so the inclusion $\mathcal{P}(\kappa )\subseteq \mathcal{P}(\lambda )$ is an injection; by the injection criterion of [F3], $\mid \mathcal{P}(\kappa )\mid \le \mid \mathcal{P}(\lambda )\mid$. Applying [F1] at $\kappa$ and at $\lambda$ turns this into $2^{\kappa }\le 2^{\lambda }$, which is clause (b).

Clause (c) is [F2] at $\kappa$: $\mathrm{cf}(2^{\kappa })>\kappa$ for every infinite cardinal $\kappa$, in particular for every infinite regular one.

Clauses (a), (b) and (c) hold for all infinite regular cardinals, so in ZFC the continuum function on infinite regular cardinals satisfies exactly the displayed constraints. The Axiom of Choice enters only through [F1] and [F2] -- through the identification of $2^{\kappa }$ with $\mid \mathcal{P}(\kappa )\mid$ and through König's theorem -- and no further selection is made in steps 1.2 and 1.3. ∎