Proper forcing preserves stationary subsets of omega-one

Choose a sufficiently large well-ordered $H_{\theta }$ structure containing $P,p,S,\dot{C}$, and fix Skolem functions for it. We first derive the elementary-model trace fact needed here. For $\beta <{\omega }_1$, let $M_{\beta }$ be the Skolem hull of $\beta \cup \{P,p,S,\dot{C}\}$ and put $f(\beta )=\sup (M_{\beta }\cap {\omega }_1)+1<{\omega }_1\mathrm{.}$ The set $E$ of nonzero limit ordinals $\delta <{\omega }_1$ closed under $f$ is club. If $\delta \in E$, finite character of Skolem terms gives $M_{\delta }={\bigcup }_{\beta <\delta }{M}_{\beta }$, so $M_{\delta }\cap {\omega }_1=\delta$. By stationarity choose $\delta \in S\cap E$ and set $M=M_{\delta }$. Then $M$ is countable elementary, contains all the required parameters, and has trace $M\cap {\omega }_1=\delta$. Properness supplies an $(M,P)$-master $q\le p$.

In $M$ choose a name $\dot{f}$ which $p$ forces to be the increasing continuous enumeration of $\dot{C}$. For every $\alpha <\delta$, one has $\alpha \in M$ and hence the ordinal name $\dot{f}(\check{\alpha })$ belongs to $M$. By F1, $q$ forces its value into $M\cap {\omega }_1=\delta$. Thus $q$ forces $\dot{f}\mathrm{‘}\mathrm{‘}\delta \subseteq \delta$. Since an increasing enumeration satisfies $\dot{f}(\alpha )\ge \alpha$, its first $\delta$ values are cofinal in $\delta$; closure of $\dot{C}$ then gives $q⊩\delta \in \dot{C}$. As $\delta \in S$ is a ground ordinal, $q⊩\dot{C}\cap \check{S}\mathrm{\ne }\mathrm{\varnothing }$.

The choices of $p$ and the club name were arbitrary, so no condition can force a ground stationary $S$ to become nonstationary. To see preservation of ${\omega }_1$ without a hidden cofinality inference, let $p$ force that $\dot{g}:\omega \to {\omega }_1^V$ is any function, choose a relevant countable model $M$ containing $p,\dot{g}$, and use properness to choose an $(M,P)$-master $q\le p$. F1 then forces each $\dot{g}(n)$ into the fixed countable ordinal $M\cap {\omega }_1$, so the range is bounded and $\dot{g}$ is not cofinal, hence not surjective. Therefore ${\omega }_1^V$ remains uncountable and equals the extension's ${\omega }_1$. AC is used exactly in A1.