. L O ] 1 5 O ct 1 99 6 Subalgebras of Cohen algebras need not be

نویسنده

  • Saharon Shelah
چکیده

Let us denote by Cκ the standard Cohen algebra of π-weight κ, i.e. the complete Boolean algebra adjoining κ Cohen reals, where κ is an infinite cardinal or 0. More generally, we call a Boolean algebra A a Cohen algebra if (for technical convenience in Theorems 0.3 and 0.4 below) it satisfies the countable chain condition and forcing with A (more precisely with the partial ordering A \ {0}) is equivalent to Cohen forcing, i.e. if every generic extension of the universe of set theory arising from forcing with A arises from forcing with some standard Cohen algebra. Since forcing with an arbitrary Boolean algebra is equivalent to forcing with its completion and forcing with a product of algebras is equivalent to forcing with one of the factors, an algebra is Cohen iff its completion is isomorphic to a product of at most countably many standard Cohen algebras; we will use this description as the definition of a Cohen algebra in the rest of the paper. Cohen algebras are among the most important objects to be studied in the realm of Boolean algebras or forcing. There is a general feeling that more or less “every” algebraic property of the standard Cohen algebras is well-known; similarly, the effect of adding Cohen reals to a given model of set theory is, generally, quite well understood. It is therefore quite surprising that the answer to an apparently innocent question was open, up to now; cf. Problem 5.2 in [Koppelberg, 1993].

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تاریخ انتشار 2008