Large model constructions for second-order ZF in dependent type theory

Dominik Kirst, Gert Smolka

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

8 Scopus citations

Abstract

We study various models of classical second-order set theories in the dependent type theory of Coq. Without logical assumptions, Aczel's sets-as-trees interpretation yields an intensional model of second-order ZF with functional replacement. Building on work of Werner and Barras, we discuss the need for quotient axioms in order to obtain extensional models with relational replacement and to construct large sets. Specifically, we show that the consistency strength of Coq extended by excluded middle and a description operator on well-founded trees allows for constructing models with exactly n Grothendieck universes for every natural number n. By a previous categoricity result based on Zermelo's embedding theorem, it follows that those models are unique up to isomorphism. Moreover, we show that the smallest universe contains exactly the hereditarily finite sets and give a concise independence proof of the foundation axiom based on permutation models.

Original languageEnglish
Title of host publicationCPP 2018 - Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with POPL 2018
EditorsAmy Felty, June Andronick
PublisherAssociation for Computing Machinery, Inc
Pages228-239
Number of pages12
ISBN (Electronic)9781450355865
DOIs
StatePublished - 8 Jan 2018
Externally publishedYes
Event7th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2018 - Los Angeles, United States
Duration: 8 Jan 20189 Jan 2018

Publication series

NameCPP 2018 - Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with POPL 2018
Volume2018-January

Conference

Conference7th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2018
Country/TerritoryUnited States
CityLos Angeles
Period8/01/189/01/18

Keywords

  • Consistency strength
  • Coq
  • Dependent type theory
  • Second-order set theory
  • Sets-as-trees model constructions

ASJC Scopus subject areas

  • Computer Science Applications
  • Software

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