Proof of Solovay's theorem

Question

I'm reading, on Jech's book Set Theory, 3rd millenium edition, the proof of Solovay's theorem for measurability of projective sets. The proof goes on using some properties of Boolean algebras.

In particular, let $M$ be the $\operatorname{ZFC}$-model $M[G]$, the generic extension carried out by the Lévy collapse $Col(\aleph_0,< \kappa)$, $\kappa \in M$ inaccessible, and $B$ the relating Boolean algebra. If $s\in M[G]$ is a countable sequence of ordinals of $M$, there exists a complete subalgebra $D\subseteq B$ such that $M[s]=M[D\cap G]$ (this happens every time i have a model between $M$ and $M[G]$, and this is the Lemma 15.43 of the book).

What I can't see (in the Lemma 26.16 in the proof of the theorem) are the following:

• Since $B$ is $\kappa$-saturated, there exists a subalgebra $D\subseteq B$ such that $|D| < \kappa$, and $M[s]=M[D\cap G]$.

and, immediately,

• It follows that $\kappa$ is inaccessible in $M[s]$.

I'd say these follow only by basic properties of Boolean algebras (and the construction of $D$ in Lemma 15.43) and by the definition of inaccessible cardinals, but I can't see why they hold.

Answer 1

1. Fix a name $\dot s$ for $s$. Fix $\eta$ large enough such that $$ \Vdash_B \dot s \subseteq \eta. $$ For each $\alpha \in \eta$ let $$ u_\alpha = \| \check{\alpha} \in \dot s \|. $$ Since $B$ is $\kappa$-saturated there is a partition $W \subseteq \{ u_\alpha \mid \alpha \in \eta \}$ of size $< \kappa$. Let $D$ be the subalgebra of $B$ generated by $W$. Since $\kappa$ is regular, we have $|D | < \kappa$. By the proof of Corollary 15.42 it furthermore follows that $$ M[s] = M[D \cap G]. $$
2. Let $\delta = |D |$. $D$ has the $\delta < \kappa$-chain condition and therefore $\kappa$ remains a regular cardinal in $M[G \cap D]$. Furthermore, for $\lambda < \kappa$, there are at most $$ (\delta^\delta)^\lambda < \kappa $$ nice $D$-names for subsets of $\lambda$. (I can't find the relevant notion in Jech's book but in Kunen's Set Theory book it's discussed on p.266.) Hence $$M[D \cap G] \models 2^{\lambda} < \kappa$$ and $\kappa$ remains a strong limit in $M[D \cap G]$.