Solovay model

A model of set theory in which every set of reals is Lebesgue measurable, constructed from an inaccessible cardinal by forcing and restricting to a definable inner model.

Forcing: A method for constructing extensions of models of set theory by adding generic sets. A forcing extension first adds reals before the Solovay model is formed.
Lebesgue measure: The translation-invariant measure on the real line that assigns intervals their length. Measurability in the model is defined relative to this standard measure.
Robert M. Solovay: An American mathematician known for foundational work in set theory and logic. Solovay introduced the model and proved its measurability result.
Nonmeasurable set: A set that cannot be assigned a measure consistently with the relevant measure axioms. Such sets exist under full choice, but none belong to the Solovay model.
Levy collapse: A forcing notion that collapses selected cardinals to smaller cardinalities. The construction collapses the inaccessible cardinal to become countable in the extension.
Axiom of choice: The principle that every family of nonempty sets has a choice function. The model omits full choice, whose consequences include nonmeasurable sets of reals.
Axiom of constructibility: The axiom asserting that every set is constructible in Gödel’s hierarchy L. Solovay’s result contrasts with the constructible universe, where choice yields nonmeasurable sets.
Vitali set: A subset of the real numbers formed by choosing one representative from each rational-translation equivalence class. The usual choice-based construction of this nonmeasurable set cannot be carried out inside the model.
Generic extension: A larger model of set theory obtained by adjoining a generic object through forcing. The initial extension supplies reals used in the model's inner construction.
Inaccessible cardinal: An uncountable regular strong limit cardinal. Its existence supplies the large-cardinal assumption used in the construction.

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