Solovay model

Solovay model The Solovay model is a model of Zermelo–Fraenkel set theory with the Axiom of Choice modified, produced by Robert M. Solovay that exhibits a universe in which every set of real numbers is Lebesgue measurable. It connects work of Paul Cohen on forcing, Kurt Gödel on constructible universes, and Dana Scott on models of set theory, and it influenced later developments by Saharon Shelah, John Steel, and W. Hugh Woodin. The construction uses an inaccessible cardinal, measure theory from Henri Lebesgue, and combinatorial techniques related to Adolf Lindenbaum and Alfred Tarski.

Background and motivation

Solovay built on independence results from Paul Cohen, Kurt Gödel, and Gerhard Gentzen by addressing measure-theoretic regularity properties using set-theoretic methods. The problem originated in questions studied by Henri Lebesgue, Émile Borel, and Felix Hausdorff concerning pathological subsets of the real line and was sharpened by results of John von Neumann and Stefan Banach about nonmeasurable sets. Prior work by Felix Hausdorff and Antoni Zygmund on descriptive set theory set the stage for Solovay's use of forcing from Paul Cohen and inner model ideas from Kurt Gödel. Solovay also drew on large cardinal concepts developed by Kurt Gödel and later formalized by William Reinhardt and Dana Scott.

Construction of the model

Solovay's construction begins in a ground model satisfying ZFC plus the existence of an inaccessible cardinal, a notion explored by Georg Cantor and axiomatized in work by Ernst Zermelo and Abraham Fraenkel. Using the forcing technique introduced by Paul Cohen, Solovay performs a Levy collapse, a specific Easton-style iteration related to methods used by William Easton and Kenneth Kunen, to make the inaccessible cardinal become \aleph_1 in the extension. The construction then takes the inner model of hereditarily definable sets of ordinals relative to the real parameters, a maneuver reminiscent of the constructible universe L developed by Kurt Gödel and elaborated by J. Barkley Rosser. Solovay draws on measure constructions comparable to those of Henri Lebesgue and measurable cardinal concepts studied by Robert Solovay's contemporaries such as Menas and Magidor.

Properties and consequences

In Solovay's model every subset of the real numbers is Lebesgue measurable, has the property of Baire, and has the perfect set property, connecting to classical results by Henri Lebesgue, Ralph Fox, and R. H. Bing. The model satisfies ZF and Dependent Choice (DC), but not the full Axiom of Choice AC, a distinction central to work by Errett Bishop and Paul Halmos. The model demonstrates independence phenomena akin to those in Paul Cohen's work on the Continuum Hypothesis and complements Gödel's constructibility results from Kurt Gödel's L. Consequences include implications for descriptive set theory pursued by Yuri M. Manin, Alexander S. Kechris, and Donald A. Martin and for determinacy studies linked to John R. Steel and W. Hugh Woodin.

Variants and generalizations

Subsequent variants replace the original inaccessible cardinal hypothesis with weaker large cardinal assumptions studied by Menas, Magidor, Solovay, and Silver. Generalizations combine Solovay-style inner model selection with forcing techniques developed by Kenneth Kunen, Saharon Shelah, and James E. Baumgartner to generate models where sets of reals satisfy various regularity properties while calibrating choice principles like the Axiom of Dependent Choice studied by H. Jerome Keisler. Modern treatments relate Solovay constructions to determinacy axioms investigated by Donald A. Martin and John R. Steel and to the fine-structural inner models advanced by Mitchell and Jensen.

Applications and impact on set theory

The Solovay model catalyzed research across descriptive set theory, forcing, and large cardinals, influencing mathematicians such as Alexander S. Kechris, Donald A. Martin, W. Hugh Woodin, Saharon Shelah, and John R. Steel. It clarified the roles of the Axiom of Choice and regularity properties, affecting subsequent work on determinacy axioms, measure theory by Henri Lebesgue, and the study of cardinal characteristics of the continuum pursued by Paul Erdős and Miklós Révész. The techniques inform modern investigations into inner model theory by William Mitchell and Ronald Jensen and into forcing axioms connected to Tomas Jech and Martin Zeman. Solovay's model remains a canonical example demonstrating how assumptions about large cardinals alter the landscape of possible regularity properties for sets of real numbers.

Category:Set theory