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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.

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    $\begingroup$ Don't all the ZFC theorems proved via forcing fit your Boolean-valued model paradigm? If not, could you clarify why not and what you want exactly? $\endgroup$ – Joel David Hamkins Feb 14 '15 at 3:23
  • $\begingroup$ Joel David Hamkins. I clarified that I want to know about combinatorial results on complete Boolean algebras. $\endgroup$ – Joseph Van Name Feb 14 '15 at 3:41
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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.

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    $\begingroup$ I don't have my copy of Jech's book handy, so the following might be in the section you cited, but if not then it could be added here: There is no free complete Boolean algebra on a countably infinite set of generators. $\endgroup$ – Andreas Blass Feb 14 '15 at 15:05
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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.

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