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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
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Proving results about complete Boolean algebras in ZFC using Boolean valued models
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 d …