Solovay model

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In the mathematical field of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, the Solovay model is a
model A model is an informative representation of an object, person or system. The term originally denoted the plan A plan is typically any diagram or list of steps with details of timing and resources, used to achieve an Goal, objective to do somet ...
constructed by in which all of the axioms of
Zermelo–Fraenkel set theory In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...
(ZF) hold, exclusive of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

, but in which all sets of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s are
Lebesgue measurable In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucli ...
. The construction relies on the existence of an
inaccessible cardinal In set theory, an uncountable set, uncountable cardinal number, cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is ...
. In this way Solovay showed that the axiom of choice is essential to the proof of the existence of a
non-measurable set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, at least granted that the existence of an inaccessible cardinal is consistent with , the axioms of Zermelo–Fraenkel set theory including the axiom of choice.

# Statement

ZF stands for Zermelo–Fraenkel set theory, and DC for the
axiom of dependent choice In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an
inner model In set theory, a branch of mathematical logic, an inner model for a Theory (mathematical logic), theory ''T'' is a substructure (mathematics), substructure of a model (mathematical logic), model ''M'' of a set theory that is both a model for ''T'' ...
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), ''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 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 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 , 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, 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 Projective hierarchy#Table, Σ 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 Σ condition is close to the best possible by constructing a model (without using an inaccessible cardinal) in which all Δ sets of reals are measurable. See and and for expositions of Shelah's result. 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

* * * * * * * * *{{Citation , last1=Stern , first1=Jacques , title=Le problème de la mesure , mr=768968 , year=1985 , journal=Astérisque , issn=0303-1179 , issue=121 , pages=325–346 Set theory Measure theory Large cardinals