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Can you take Solovay’s inaccessible away? (English) Zbl 0596.03055

In this paper the author solves a long-standing, famous, open question of set theory showing that if every set of reals is Lebesgue measurable, then (the real) \(\aleph_ 1\) is an inaccessible cardinal in L \((ZF+DC\) is used). This is a counterpart of Solovay’s famous theorem [R. M. Solovay, Ann. Math., II. Ser. 92, 1-56 (1970; Zbl 0207.009)]. It is proved that if there exists a set of \(\aleph_ 1\) reals, then there is a non-measurable set, and if \(\aleph_ 1\) is not inaccessible in L, then there is a non-measurable \(\Sigma^ 1_ 3\) set. This is sharp in the sense that every model of set theory has a forcing extension in which all \(\Delta^ 1_ 3\) sets are measurable. Also, that every set of reals has the Baire property is outright consistent with \(AF+DC\), and it is even possible to insure that if \(\{A_ x:\) \(x\in R\}\) is a collection of non- empty sets of reals, indexed by reals, then a certain real function is a choice function except for a first-category set. For this, the author introduces the notion of sweet forcing, and proves a series of technical lemmas.
Some other, loosely connected, results are also included into the paper. It is proved, that when adding a Cohen-real, a Suslin tree is produced, and even very general principles will hold, which follow from \(\diamond\) and are easy to be added by finite-part forcing. If, however, a random real is added to a model of Martin’s axiom, then there are no almost- disjoint sets \(\{A_{\alpha}:\) \(\alpha <\omega_ 1\}\) in \([\omega]^{\omega}\) such that among any \(\aleph_ 1\) of them, there are three, forming a delta-system. Recent work of R. Laver shows that there are not even Suslin trees there.
In the paper reviewed below (see Zbl 0596.03056) a more elementary proof is given to the main result of the present paper.
Reviewer: P.Komjáth

MSC:

03E15 Descriptive set theory
03E25 Axiom of choice and related propositions
03E35 Consistency and independence results
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References:

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