Pointwise Definable Models of Set Theory

TL;DR

This paper explores the existence and construction of pointwise definable models of set theory, showing that many such models can be obtained from countable models through class forcing extensions, with all objects definable without parameters.

Contribution

It proves that every countable model of G"odel-Bernays set theory has a pointwise definable extension where all sets and classes are parameter-free definable.

Findings

Existence of continuum many pointwise definable models of ZFC.

Every countable model of G"odel-Bernays set theory has a pointwise definable extension.

Construction of such models via class forcing extensions.

Abstract

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Godel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.