Editing Vitali set (section)

== Role of the axiom of choice ==
The construction of Vitali sets given above uses the [[axiom of choice]]. The question arises: is the axiom of choice needed to prove the existence of sets that are not Lebesgue measurable? The answer is yes, provided that [[inaccessible cardinal]]s are consistent with the most common axiomatization of set theory, so-called [[ZFC]].

In 1964, [[Robert M. Solovay|Robert Solovay]] constructed a model of Zermelo–Fraenkel set theory without the axiom of choice where all sets of real numbers are Lebesgue measurable. This is known as the [[Solovay model]].<ref>{{Citation |last1=Solovay |first1=Robert M. |title=A model of set-theory in which every set of reals is Lebesgue measurable |journal=[[Annals of Mathematics]] |volume=92 |issue=1 |pages=1–56 |year=1970 |series=Second Series |doi=10.2307/1970696 |issn=0003-486X |jstor=1970696 |mr=0265151 |author1-link=Robert M. Solovay}}</ref> In his proof, Solovay assumed that the existence of inaccessible cardinals is [[Consistency|consistent]] with the other axioms of Zermelo-Fraenkel set theory, i.e. that it creates no contradictions. This assumption is widely believed to be true by set theorists, but it cannot be proven in ZFC alone.<ref name="wagon">{{Cite book |last1=Wagon |first1=Stan |title=The Banach-Tarski Paradox |last2=Tomkowicz |first2=Grzegorz |publisher=Cambridge University Press |year=2016 |edition=2nd |pages=296–299}}</ref>

In 1980, [[Saharon Shelah]] proved that it is not possible to establish Solovay's result without his assumption on inaccessible cardinals.<ref name=wagon/>