Solovay model
Encyclopedia
In the mathematical field of set theory
, the Solovay model is a model
constructed by 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 number
s are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal
.
In this way Solovay showed that the axiom of choice is essential to the proof of the existence of a non-measurable set
, at least granted that the existence of an inaccessible cardinal is consistent with ZFC, the axioms of Zermelo–Fraenkel set theory including the axiom of 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.
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 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.
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 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 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.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, the Solovay model is a model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
constructed by in which all of the axioms of Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
(ZF) hold, exclusive of the axiom of choice, but in which all sets of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal
Inaccessible cardinal
In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable...
.
In this way Solovay showed that the axiom of choice is essential to the proof of the existence of a non-measurable set
Non-measurable set
In mathematics, a non-measurable set is a set whose structure is so complicated that it cannot be assigned any meaningful measure. Such sets are constructed to shed light on the notions of length, area and volume in formal set theory....
, at least granted that the existence of an inaccessible cardinal is consistent with ZFC, 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 choiceAxiom of dependent choice
In mathematics, the axiom of dependent choices, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis...
.
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
Perfect set property
In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset...
, 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
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
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 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 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. 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 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 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.