Orientation for intermediate models and complete subalgebras

Let M be a transitive ZF ground model. Suppose $B\subseteq C$ are nontrivial complete Boolean algebras in M and the inclusion $B\hookrightarrow C$ is a Boolean embedding preserving all joins computed in M. Thus it preserves zero, one, complements, finite meets and finite joins as well as those ground-model joins. This is the complete-subalgebra convention: completeness refers to ground subsets, not all external subsets. See Completeness, regular opens, and order continuity. If H is M-generic for $C^{+}=C\setminus \{0\}$, then $G=H\cap B$ is M-generic for $B^{+}$ and

$M\subseteq M[G]\subseteq M[H]\mathrm{.}$

Here is the full argument for this direction. A Boolean forcing filter contains 1. If $b_0,b_1\in H\cap B$, directedness gives $c\in H$ below both. Their Boolean meet is above c, hence belongs to H by upward closure; it is nonzero and belongs to B. This proves directedness of G; nonemptiness and upward closure follow as well.

For any ground dense $D\subseteq B^{+}$, its join in B is 1: otherwise the nonzero complement of that join has a refinement $d\in D$, which would lie below both the join and its complement. The complete inclusion therefore makes its join in C also 1. For each nonzero $c\in C$, some $d\in D$ has $c\wedge d\mathrm{\ne }0$. If all those meets were zero, c would lie below every $\mathrm{\neg }d$, hence below the complement of their join, namely zero. Thus $E=\{e\in C^{+}:\mathrm{\exists }d\in D(e\le d)\}$ is a ground dense set in $C^{+}$. H meets E, and upward closure puts the corresponding d in H. So G meets D, proving genericity without any maximal-antichain selection or AC.

Each $B^{+}$-name is also a $C^{+}$-name. Induction on its subname relation shows that its value by G equals its value by H: every coefficient belongs to B, and therefore belongs to G exactly when it belongs to H; the induction hypothesis identifies all selected subname values. The definition Valuation of names and M[G] now gives $M[G]\subseteq M[H]$. The ground inclusion and ZF model assertions follow from Generic extensions satisfy ZF and preserve ground-model Choice, using its choice-free branch. If M satisfies AC, its separately qualified branch gives ZFC for both extensions. A complete subalgebra equal to C gives G=H; the two-element subalgebra gives the trivial generic and intermediate model M.

The converse claim that every intermediate model arises from a complete subalgebra is not asserted here. It requires a different theorem. In particular this argument does not infer that the inclusion $B^{+}\to C^{+}$ has dense range.