Pointwise definable models of set theory

DOI10.2178/JSL.7801090zbMATH Open1270.03101arXiv1105.4597OpenAlexW2113840635WikidataQ55896321 ScholiaQ55896321MaRDI QIDQ4916549

David Linetsky, Jonas Reitz, Joel David Hamkins

Publication date: 23 April 2013

Published in: Journal of Symbolic Logic (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1105.4597

zbMATH Keywords

countable modelGödel-Bernays set theorytransitive modelfirst-order definableclass forcing extensionpointwise definable model

Mathematics Subject Classification ID

Consistency and independence results (03E35) Inner models, including constructibility, ordinal definability, and core models (03E45) Large cardinals (03E55)

Cites Work

• Title not available (Why is that?)
• Models of set theory with definable ordinals
• Large cardinals and definable well-orders on the universe
• Some applications of Jensen's coding theorem
• Counting models of set theory
• The Hypothesis that all Classes are Nameable

Related Items (16)

Title not available (Why is that?) ⋮ Algebraicity and implicit definability in set theory ⋮ Typicality à la Russell in set theory ⋮ Concrete Models of Set Theory ⋮ Universal models and definability ⋮ On the Axiom of Canonicity ⋮ When does every definable nonempty set have a definable element? ⋮ VARIETIES OF CLASS-THEORETIC POTENTIALISM ⋮ A classical way forward for the regularity and normalization problems ⋮ TOWARDS A THEORY OF DEFINABLE SETS ⋮ Non-pointed strongly protomodular theories ⋮ Ehrenfeucht's lemma in set theory ⋮ Strong reducibilities and set theory ⋮ MINIMUM MODELS OF SECOND-ORDER SET THEORIES ⋮ Undefinable Classes and Definable Elements in Models of Set Theory and Arithmetic ⋮ f-Model Generated by P-Set

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