Editing Non-measurable set (section)

{{Short description|Set which cannot be assigned a meaningful "volume"}}
{{More citations needed|date=August 2009}}

In [[mathematics]], a '''non-measurable set''' is a [[Set (mathematics)|set]] which cannot be assigned a meaningful "volume".  The existence of such sets is construed to provide information about the notions of [[length]], [[area]] and [[volume]] in formal set theory. In [[Zermelo–Fraenkel set theory]], the [[axiom of choice]] entails that non-measurable subsets of <math>\mathbb{R}</math> exist.

The notion of a non-measurable set has been a source of great controversy since its introduction. Historically, this led [[Émile Borel|Borel]] and [[Kolmogorov]] to formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections of intervals (called [[Borel set]]s) plus-minus [[null set]]s. These sets are rich enough to include every conceivable definition of a set that arises in standard mathematics, but they require a lot of formalism to prove that sets are measurable.

In 1970, [[Robert M. Solovay]] constructed the [[Solovay model]], which shows that it is consistent with standard set theory without uncountable choice,  that all subsets of the reals are measurable. However, Solovay's result depends on the existence of an [[inaccessible cardinal]], whose existence and consistency cannot be proved within standard set theory.