Solovay's Theorem, published by Robert Solovay in 1970, shows that if the existence of an inaccessible cardinal is consistent with the Zermelo-Fraenkel axioms, then it is also consistent, without the axiom of choice, that every set of real numbers is Lebesgue measurable, has the Baire property, and has the perfect set property. The result, built from what is now called the Solovay model, established that some form of the axiom of choice is genuinely necessary to construct a non-measurable set of reals, since removing choice can make every set of reals well behaved provided a suitable large cardinal is assumed.
Facts
StatementAssuming the existence of an inaccessible cardinal, there is an inner model of ZF plus the axiom of dependent choice, built inside a suitable forcing extension, in which every set of real numbers is Lebesgue measurable, has the perfect set property, and has the Baire property. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Solovay model, Wikipedia
Sources
1. Solovay model, Wikipedia
Lead section
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.
In Branch: Set Theory, Lead sentence
In the mathematical field of set theory, the Solovay model is a model constructed by Robert M.
View the Source2. Robert M. Solovay, Wikipedia
References section, Solovay 1970 citationQuote, References section, Solovay 1970 citation
Solovay, Robert M. (1970). A model of set-theory in which every set of reals is Lebesgue measurable. Annals of Mathematics. Second Series. 92 (1): 1-56.
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