Easton's theorem for regular cardinals

By [F1] $M[G]\models \mathrm{Z}\mathrm{F}\mathrm{C}$ and $M\subseteq M[G]$ have the same ordinals; fix an infinite regular $\kappa$ of $M$. The head $P^{\le \kappa }$ is a set-sized Easton product with the ${\kappa }^{+}$-chain condition and the tail $P^{>\kappa }$ is ${\kappa }^{+}$-closed [F2]. Any subset of $\kappa$ in $M[G]$ has a name in a set stage $M[G^{\le \gamma }]$ for some infinite regular $\gamma \ge \kappa$ [F1], and the factorization $P^{\le \gamma }\cong P^{\le \kappa }\times P^{(\kappa ,\gamma ]}$ has ${\kappa }^{+}$-closed second factor and ${\kappa }^{+}$-cc first factor, so [F3] puts the subset already in $M[G^{\le \kappa }]$. The set-sized realization theorem gives $2^{\kappa }=F(\kappa )$ in $M[G^{\le \kappa }]$ [F5]. Thus the full extension has the same set of subsets of $\kappa$ as that head extension.

Every ground regular cardinal remains regular in $M[G]$: suppose $\kappa$ is infinite and regular in $M$ but not in $M[G]$, and let $\delta ={\mathrm{cf}}^{M[G]}(\kappa )<\kappa$ with a cofinal $f:\delta \to \kappa$ in $M[G]$. Then $\delta$ is regular in $M[G]$ and therefore in $M$, since otherwise a ground cofinal map of shorter length would persist into $M[G]$; so $\delta$ is an infinite regular cardinal of $M$ with $\delta <\kappa$. By [F1] the function $f$ lies in some stage $M[G^{\le \gamma }]$ with $\gamma \ge \delta$, and the factorization $P^{\le \gamma }\cong P^{\le \delta }\times P^{(\delta ,\gamma ]}$ has ${\delta }^{+}$-closed second factor and ${\delta }^{+}$-cc first factor [F2], so [F3] gives $f\in M[G^{\le \delta }]$ and hence ${\mathrm{cf}}^{M[G^{\le \delta }]}(\kappa )\le \delta$; but [F4] at $\theta ={\delta }^{+}\le \kappa$ gives ${\mathrm{cf}}^{M[G^{\le \delta }]}(\kappa )={\mathrm{cf}}^M(\kappa )=\kappa >\delta$, a contradiction.

Every ground cardinal remains a cardinal of $M[G]$: suppose $\kappa$ is the least ground cardinal with $\mid \kappa {\mid }^{M[G]}=\mu <\kappa$. By step 1.2, $\kappa$ cannot be regular in $M$, since an ordinal that remains regular in the ZFC extension is a cardinal there. Thus $\kappa$ is a singular ground cardinal, hence a limit cardinal; the ground cardinals below $\kappa$ are cofinal in $\kappa$. Choose a ground cardinal $\nu$ with $\mu <\nu <\kappa$. Minimality of $\kappa$ makes $\nu$ a cardinal in $M[G]$, whereas a bijection $\kappa \to \mu$ in $M[G]$ restricts to an injection $\nu \to \mu$, a contradiction. Hence all ground cardinals remain cardinals.

All ground cofinalities are preserved. Let $\kappa ={\mathrm{cf}}^M(\delta )$ for an ordinal $\delta$ of $M$ and fix a strictly increasing cofinal $g:\kappa \to \delta$ in $M$; then ${\mathrm{cf}}^{M[G]}(\delta )\le \kappa$ because $g$ is still cofinal in $M[G]$. If $\eta ={\mathrm{cf}}^{M[G]}(\delta )<\kappa$, take a cofinal $h:\eta \to \delta$ in $M[G]$ and define $h^{\mathrm{\prime }}:\eta \to \kappa$ in $M[G]$ by letting $h^{\mathrm{\prime }}(\xi )$ be the least $\beta <\kappa$ with $h(\xi )\le g(\beta )$; then $h^{\mathrm{\prime }}$ is cofinal in $\kappa$, because for any ${\beta }_0<\kappa$ cofinality of $h$ gives $\xi <\eta$ with $h(\xi )\ge g({\beta }_0)$, hence $g(h^{\mathrm{\prime }}(\xi ))\ge h(\xi )\ge g({\beta }_0)$ and $h^{\mathrm{\prime }}(\xi )\ge {\beta }_0$ by strict increase of $g$. So ${\mathrm{cf}}^{M[G]}(\kappa )\le \eta <\kappa$, contradicting step 1.2 because $\kappa$ is regular in $M$; therefore ${\mathrm{cf}}^{M[G]}(\delta )={\mathrm{cf}}^M(\delta )$.

By step 1.1 the full extension and the head extension have the same subsets of each regular $\kappa$, and by step 2.1 $F(\kappa )$ remains a cardinal; hence $(2^{\kappa }{)}^{M[G]}=F(\kappa )$. Steps 2.1 and 2.2 therefore give a ZFC model $M[G]\supseteq M$ with the ordinals, cardinals and cofinalities of $M$ and with $2^{\kappa }=F(\kappa )$ at every infinite regular cardinal; for the final clause, if $F$ satisfies the three necessary conditions of [F6] then $F$ is an Easton function in the sense of [F7] and the construction above realizes it, while conversely those necessary conditions must hold of $2^{\kappa }$ by [F6]; the relative-consistency reading is the one of [F8]: a constructible GCH ground with its closure classes supplies the ground, so no more than the necessary conditions is required. This is the statement. ∎