Set-sized Easton forcing preserves cardinals and cofinalities

Fix the data of the statement, so that $M\subseteq M[G]$ are transitive models with the same ordinals and $M[G]\models \mathrm{Z}\mathrm{F}$, and let $\delta \in M$ be an ordinal.

For every infinite ground regular cardinal $\lambda$ the factorization $P(F)\cong P^{\le \lambda }\times P^{>\lambda }$ holds in $M$. The coordinate projections $G^{>\lambda }$ and $G^{\le \lambda }$ of $G$ are $M$-generic: if $D\subseteq P^{\le \lambda }$ is dense and $D\in M$, then the set of conditions $p\in P(F)$ with $p^{\le \lambda }\in D$ is dense in $P(F)$ and lies in $M$, so $G$ meets it and $G^{\le \lambda }$ meets $D$; the same computation with a dense $D\subseteq P^{>\lambda }$ handles the tail. Hence $M[G]=M[G^{>\lambda }][G^{\le \lambda }]$.

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 an infinite regular cardinal of $M[G]$ and hence also of $M$: if ${\mathrm{cf}}^M(\delta )<\delta$ then $M[G]$ would contain a cofinal map ${\mathrm{cf}}^M(\delta )\to \delta$ by [F7], making $\delta$ singular in $M[G]$. So $\delta$ is an infinite regular cardinal of $M$ with $\delta <\kappa$, and by step 1.2 the lemma [F4] applies with the ${\delta }^{+}$-closed tail $P^{>\delta }$ and the ${\delta }^{+}$-cc head $P^{\le \delta }$, giving $f\in M[G^{\le \delta }]$. Thus ${\mathrm{cf}}^{M[G^{\le \delta }]}(\kappa )\le \delta$, while [F5] 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 not, and let $\kappa$ be the least ground cardinal with $\mid \kappa {\mid }^{M[G]}=\mu <\kappa$. By step 2.1, $\kappa$ cannot be regular in $M$: an ordinal regular in the ZFC extension is a cardinal there. Thus $\kappa$ is a singular ground cardinal, hence a limit cardinal, and the ground cardinals below it are cofinal in it. Choose a ground cardinal $\nu$ with $\mu <\nu <\kappa$. Minimality of $\kappa$ makes $\nu$ a cardinal of $M[G]$. A bijection $\kappa \to \mu$ in $M[G]$ restricts to an injection $\nu \to \mu$, since $\nu \subseteq \kappa$ and $\nu$ is an ordinal of $M[G]$; hence $\mid \nu {\mid }^{M[G]}\le \mu <\nu$, a contradiction.

All ground cofinalities are preserved. Let $\kappa ={\mathrm{cf}}^M(\delta )$ and fix a strictly increasing cofinal $g:\kappa \to \delta$ in $M$, so that ${\mathrm{cf}}^{M[G]}(\delta )\le \kappa$ because $g$ still has unbounded range 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 $h^{\mathrm{\prime }}$ has cofinal range in $\kappa$: given ${\beta }_0<\kappa$, cofinality of $h$ gives $\xi$ 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 ${\mathrm{cf}}^{M[G]}(\kappa )\le \eta <\kappa$ would hold, contradicting step 2.1 since $\kappa$ is ground regular; therefore ${\mathrm{cf}}^{M[G]}(\delta )=\kappa ={\mathrm{cf}}^M(\delta )$.

Steps 3.2 and 3.1 show that $M$ and $M[G]$ have the same ordinals, the same cofinality function on the ordinals of $M$, and the same cardinals, which is the statement. ∎