A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable

1970/07/01 by Robert M Solovay · 22 citations
Mathematics · Decision Sciences · #Advanced Topology and Set Theory #Decision-Making and Behavioral Economics

paper · doi:10.2307/1970696

openalex publication_date 1970/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/03

Abstract

We show that the existence of a non-Lebesgue measurable set cannot be proved in Zermelo-Frankel set theory (ZF) if use of the axiom of choice is disallowed. In fact, even adjoining an axiom DC to ZF, which allows countably many consecutive choices, does not create a theory strong enough to construct a non-measurable set. Let ZFC be Zermelo-Frankel set theory together with the axiom of choice. Let I be the statement: There is an inaccessible cardinal'.

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