= Solution
The <Rasiowa–Sikorski lemma> constructs a <generic filter> $G_i$ over each $M_i$. Every resulting <generic extension> remains a <countable transitive model> of <ZFC>: there are only countably many ground-model names externally. The order $\mathbb P$ has the same elements and ordering at every stage, and <atomless forcing order> structure is absolute because all its quantifiers are bounded to $\mathbb P$. Hence the <generic filter for an atomless order is new> result gives $G_i\notin M_i$ for every $i$.
The increasing union $N=\bigcup_{i<\omega}M_i$ is transitive and contains $\mathbb P$. If it satisfied <Axiom of power set> for $\mathbb P$, there would be a <set> $A\in N$ with
$$
N\models\forall z\,(z\subseteq\mathbb P\leftrightarrow z\in A).
$$
Choose $i$ with $A\in M_i$. The next-stage <generic filter> $G_i$ belongs to $M_{i+1}\subseteq N$ and is an actual <subset> of $\mathbb P$. This subset assertion is absolute for the <transitive set> $N$, so $G_i\in A$. Transitivity of $M_i$ and $A\in M_i$ then give $G_i\in M_i$, a contradiction. Thus the <power-set failure in an increasing union of generic extensions> occurs already at the fixed ground-model order:
$$
\boxed{N\not\models\mathsf{PowerSet}\quad\text{because no }\mathcal P^N(\mathbb P)\text{ belongs to }N.}
$$
The stages form an increasing chain, not an elementary chain, so the <elementary chain theorem> does not assert <ZFC> for their union.
Back to article page
