Solovay model

In the mathematical field of set theory, the Solovay model is a model constructed by Robert M. Solovay (1970) in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal. In this way Solovay showed that in the proof of the existence of a non-measurable set from ZFC (Zermelo–Fraenkel set theory plus the axiom of choice), the axiom of choice is essential, at least granted that the existence of an inaccessible cardinal is consistent with ZFC.

Statement

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.

Construction

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.

Complements

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 Σ13 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 Σ13 condition is close to the best possible by constructing a model (without using an inaccessible cardinal) in which all Δ13 sets of reals are measurable. See (Raisonnier 1984) and (Stern 1985) and (Miller 1989) for expositions of Shelah's result.

(Shelah Woodin) 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.

References

• Krivine, Jean-Louis (1969), "Modèles de ZF + AC dans lesquels tout ensemble de réels définissable en termes d'ordinaux est mesurable-Lebesgue", Comptes Rendus de l'Académie des Sciences, Série A et B 269: A549–A552, ISSN 0151-0509 
• Krivine, Jean-Louis (1971), "Théorèmes de consistance en théorie de la mesure de R. Solovay", Séminaire Bourbaki vol. 1968/69 Exposés 347-363, Lecture Notes in Mathematics, 179, pp. 187–197, doi:10.1007/BFb0058812, ISBN 978-3-540-05356-9, http://www.numdam.org/item/SB_1968-1969__11__187_0/ 
• Miller, Arnold W. (1989), "Review of "Can You Take Solovay's Inaccessible Away? by Saharon Shelah"", The Journal of Symbolic Logic (Association for Symbolic Logic) 54 (2): 633–635, doi:10.2307/2274892, ISSN 0022-4812 
• Raisonnier, Jean (1984), "A mathematical proof of S. Shelah's theorem on the measure problem and related results.", Israel Journal of Mathematics 48: 48–56, doi:10.1007/BF02760523 
• Shelah, Saharon (1984), "Can you take Solovay's inaccessible away?", Israel Journal of Mathematics 48 (1): 1–47, doi:10.1007/BF02760522, ISSN 0021-2172 
• Shelah, Saharon; Woodin, Hugh (1990), "Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable", Israel Journal of Mathematics 70 (3): 381–394, doi:10.1007/BF02801471, ISSN 0021-2172 
• Solovay, Robert M. (1970), "A model of set-theory in which every set of reals is Lebesgue measurable", Annals of Mathematics, Second Series 92 (1): 1–56, doi:10.2307/1970696, ISSN 0003-486X 
• Specker, Ernst (1957), "Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom)", Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 3 (13–20): 173–210, doi:10.1002/malq.19570031302, ISSN 0044-3050 
• Stern, Jacques (1985), "Le problème de la mesure", Astérisque (121): 325–346, ISSN 0303-1179