Proving results about complete Boolean algebras in ZFC using Boolean valued models

Question

I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing. I am mainly interested in proofs of combinatorial properties of complete Boolean algebras such as the many cardinal invariants on Boolean algebras. The proofs of results that I have in mind have the following form.

$\bullet$ One translates combinatorial properties of complete Boolean algebras $B$ to properties of the Boolean valued models $V^{B}$.

$\bullet$ One proves things about the Boolean valued models $V^{B}$.

$\bullet$ One translates the results about the Boolean valued models $V^{B}$ back to results about the complete Boolean algebras $B$.

I am interested in proofs of results where the only known proof uses forcing or where the proof without forcing is more difficult or at least equally difficult.

This question and this question ask about ZFC results that can be proven using forcing, but now I am only asking about combinatorial results about complete Boolean algebras proven using forcing using the strategy mentioned above.

Answer 1

Let me say a few such examples:

1) (Kripke's theorem): For every Boolean lagebra $B$, there is a cardinal $\kappa$ such that $B$ can be embedded in the collapsing algebra $Col(\aleph_0, \kappa).$

2) (Solovay's theorem): Let $B$ be a Souslin algebra. Then $|B|\leq 2^{\aleph_1}$ (see Jech 1978, Theorem 60, page 274).

In fact most of section 25 ``Forcing and complete Boolean algebras'' of Jech 1978, gives such examples.

3) (Jensen's theorem): Let $\kappa$ be an inaccessible cardinals and let $B=RO(P),$ where $P=Col(\aleph_0, < \kappa).$ If $B_1, B_2$ are complete Boolean algebras of size $<\kappa$ such that $B_1$ is a sub-algebra of both $B$ and $B_1$, then there is an embedding $h$ of $B_2$ into $B$ such that $h\restriction B_1$ is the identity.

Answer 2

You can, for example, show that many standard complete boolean algebras like Cohen algebra cannot be isomorphic to a quotient $\mathcal{P}(X) / I$ for any sigma ideal $I$ over $X$. These results are due to Gitik and Shelah and their proofs use generic ultrapowers.