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arXiv:1105.4597math.LO

Pointwise Definable Models of Set Theory

Joel David Hamkins, David Linetsky, Jonas Reitz

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.

arXiv comments
23 pages
Journal reference
DOI
Subjects
math.LOLogic

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  1. Why Some Infinities Are Bigger Than Others 1 comment

    • A fun wrinkle is that there are (infinitely many) models of set theory which are pointwise definable. Thus in the right model of zfc, every mathematical object is uniquely addressable by a certain finite formula. (see https://arxiv.org/pdf/1105.4597.pdf )

      chobytesView on HNParent context

  2. How real are real numbers? (2004) 1 comment

    • A rather more technical – but very interesting – take on this question is [0], which opens: > One occasionally hears the argument — let us call it the math-tea argument, for perhaps it is heard at a good math tea — that there must be real numbers that we cannot describe or…

      robinhoustonView on HNParent context

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