A characterization of permutation models in terms of forcing
type: Item
instance of: scholarly article
title: A characterization of permutation models in terms of forcing
zbMATH Open document ID: 1047.03037
author: Eric J. Hall
published in: Notre Dame Journal of Formal Logic
publication date: 2004-05-27T00:00:00Z
review text: ZFA denotes ZF set theory altered to allow a set of atoms. A known phenomenon is the existence of statements that are independent of ZFA but yet hold in every permutation model. The author produces a sentence of set theory \(\psi\) that is true in all permutation models and has the property that if \(\phi\) is any sentence that is true in every permutation model, then ZFA\( + \psi \vdash \phi\). The existence of \(\psi\) follows from the author's main theorem, which in non-technical terms asserts that a transitive model of ZFA is a permutation model if and only if some generic extension of the transitive model with the same kernel satisfies AC. The sentence \(\psi\) is obtained from this characterization because forcing is definable in set theory.
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 2067617
zbMATH Keywords: axiom of choice
zbMATH Keywords: ZFA
zbMATH Keywords: permutation model
reviewed by: J. M. Plotkin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1074290714
OpenAlex ID: W1994055707
cites work: Injectivity, Projectivity, and the Axiom of Choice
cites work: Freyd’s models for the independence of the axiom of choice
cites work: The Fraenkel-Mostowski method, revisited
cites work: Permutation models and topological groups
cites work: Intermediate submodels and generic extensions in set theory
cites work: Limitations on the Fraenkel-Mostowski method of independence proofs
cites work: The axiom of choice
DOI: 10.1305/NDJFL/1074290714
description: scientific article; zbMATH DE number 2067617