= Solution
Let $\kappa=(\omega_2)^M$ be the fixed ground <ordinal>. We prove that $S$ remains stationary in this <ordinal>. The <chain condition for forcing> ensures that $\kappa$ remains regular: possible values of each coordinate of an ordinal-valued name form a ground set of size less than $\kappa$, by a maximal deciding <forcing antichain>. A hypothetical cofinal map with domain below $\kappa$ would have its range covered by fewer than $\kappa$ such small sets, hence bounded by regularity in $M$.
Let $\dot C$ name a <club set> in $\kappa$, and take $p\in G$ <forcing> this. For each $\alpha<\kappa$, choose in $M$ a maximal <forcing antichain> above $p$ deciding the least point of $\dot C$ strictly above $\alpha$. Its set of possible values has size less than $\kappa$, so choose a ground bound $h(\alpha)<\kappa$ larger than all of them. The ground set
$$
D=\{\delta<\kappa:\delta\text{ is limit and }(\forall\alpha<\delta)\ h(\alpha)<\delta\}
$$
is a <club set> by the usual countable closure iteration and regularity. For every $\delta\in D$, $p$ forces $\dot C\cap\delta$ unbounded in $\delta$, and closedness forces $\delta\in\dot C$. Hence $p\Vdash\check D\subseteq\dot C$. In $M$, choose $\delta\in S\cap D$; that same <ordinal> belongs to $S\cap C$ in the extension. \b[Stationarity at the fixed ground $\kappa$ is preserved.]
There is an important qualification to the printed $\omega_2$ notation. If it is recomputed internally as $(\omega_2)^{M[G]}$, the assertion is false without preservation of smaller <cardinals>. Finite partial maps from $\omega$ to $(\omega_1)^M$ form a <forcing> of size $(\aleph_1)^M$, hence have the $(\aleph_2)^M$-chain condition, but collapse $(\omega_1)^M$ to countable. Then $(\omega_2)^M=(\omega_1)^{M[G]}$, while $(\omega_2)^{M[G]}$ is larger. The old $S$ is bounded in this new $\omega_2$, so cannot be stationary there. The proved statement uses the fixed ground <ordinal>, or alternatively requires the lower-cardinal preservation needed to retain its aleph index.
Back to article page
