Pointwise definable models of set theory

E-ISSN： 1943-5886|78|1|139-156

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.78, Iss.1, 2013-03, pp. : 139-156

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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 Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.