In the mathematical field of
set theory
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, set theory, as a branch of mathematics, is mostly conce ...
, the Solovay model is a
model
A model is an informative representation of an object, person or system. The term originally denoted the Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...
constructed by in which all of the axioms of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
(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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
, but in which all
sets of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s are
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
. The construction relies on the existence of an
inaccessible cardinal
In set theory, an uncountable 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 uncountable, it is not a sum of ...
.
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, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zerm ...
, 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 choice In mathematics, the axiom of dependent choice, denoted by \mathsf , is a weak form of the axiom of choice ( \mathsf ) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores whi ...
.
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 ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let L = \langle \in \rangle be ...
of ZF + DC of a suitable
forcing extension ''V''
'G''such that every set of reals is Lebesgue measurable, has the
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 (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a p ...
, and has the
Baire property
A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such t ...
.
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 In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963.
The collapsing alge ...
''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 set
In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is
* \boldsymbol^1_1 if A is analytic
* \boldsymbol^1_n if the complement of A, X\se ...
s of reals; however for reasons related to
Tarski's undefinability theorem
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that ''arithmetical truth ...
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 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 ℵ
1 is regular then ℵ
1 is inaccessible in the
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
. 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 ℵ
1 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 cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties.
Formal definition
If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...
s 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
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*{{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