Solovay model

ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice.

Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension V[G] such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property.

Solovay constructed his model in two steps, starting with a model M of ZFC containing an inaccessible cardinal κ.

The first step is to take a Levy collapse M[G] of M by adding a generic set G for the notion of forcing that collapses all cardinals less than κ to ω. Then M[G] is a model of ZFC with the property that every set of reals that is definable over a countable sequence of ordinals is Lebesgue measurable, and has the Baire and perfect set properties. (This includes all definable and projective sets of reals; however for reasons related to Tarski's undefinability theorem the notion of a definable set of reals cannot be defined in the language of set theory, while the notion of a set of reals definable over a countable sequence of ordinals can be.)

The second step is to construct Solovay's model N as the class of all sets in M[G] that are hereditarily definable over a countable sequence of ordinals. The model N is an inner model of M[G] satisfying ZF + DC such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property. The proof of this uses the fact that every real in M[G] is definable over a countable sequence of ordinals, and hence N and M[G] have the same reals.

Instead of using Solovay's model N, one can also use the smaller inner model L(R) of M[G], consisting of the constructible closure of the real numbers, which has similar properties.

Solovay suggested in his paper that the use of an inaccessible cardinal might not be necessary. Several authors proved weaker versions of Solovay's result without assuming the existence of an inaccessible cardinal. In particular Krivine (1969) showed there was a model of ZFC in which every ordinal-definable set of reals is measurable, Solovay showed there is a model of ZF + DC in which there is some translation-invariant extension of Lebesgue measure to all subsets of the reals, and Shelah (1984) showed that there is a model in which all sets of reals have the Baire property (so that the inaccessible cardinal is indeed unnecessary in this case).

The case of the perfect set property was solved by Specker (1957), who showed (in ZF) that if every set of reals has the perfect set property and the first uncountable cardinal ℵ₁ is regular then ℵ₁ is inaccessible in the constructible universe. Combined with Solovay's result, this shows that the statements "There is an inaccessible cardinal" and "Every set of reals has the perfect set property" are equiconsistent over ZF.

Finally, Shelah (1984) showed that consistency of an inaccessible cardinal is also necessary for constructing a model in which all sets of reals are Lebesgue measurable. More precisely he showed that if every Σ¹
₃ set of reals is measurable then the first uncountable cardinal ℵ₁ is inaccessible in the constructible universe, so that the condition about an inaccessible cardinal cannot be dropped from Solovay's theorem. Shelah also showed that the Σ¹
₃ condition is close to the best possible by constructing a model (without using an inaccessible cardinal) in which all Δ¹
₃ sets of reals are measurable. See Raisonnier (1984) and Stern (1985) and Miller (1989) for expositions of Shelah's result.

Shelah & Woodin (1990) showed that if supercompact cardinals exist then every set of reals in L(R), the constructible sets generated by the reals, is Lebesgue measurable and has the Baire property; this includes every "reasonably definable" set of reals.

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