= Power-set failure in an increasing union of generic extensions
Let $M_{n+1}=M_n[G_n]$ for a fixed <atomless forcing order> $\mathbb P$, and $N=\bigcup_{n<\omega}M_n$. If a <set> $A\in N$ were its internal <power set> of $\mathbb P$, choose $n$ with $A\in M_n$. The next <generic filter> $G_n$ belongs to $N$, so $G_n\in A$; transitivity of $M_n$ implies $G_n\in M_n$, contradicting <generic filter for an atomless order is new>. The chain is increasing but need not be an elementary chain.
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