Pointwise Definable Models of Set Theory
Logic
2012-06-20 v2
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.
Keywords
Cite
@article{arxiv.1105.4597,
title = {Pointwise Definable Models of Set Theory},
author = {Joel David Hamkins and David Linetsky and Jonas Reitz},
journal= {arXiv preprint arXiv:1105.4597},
year = {2012}
}
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23 pages