A category-theoretic approach to Boolean-valued models of set theory
type: Item
instance of: scholarly article
title: A category-theoretic approach to Boolean-valued models of set theory
zbMATH Open document ID: 0626.03046
DOI: 10.1016/0022-4049(86)90010-1
author: G. P. Monro
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \({\mathbb{B}}\) be a complete Boolean algebra. Scott and Solovay constructed a \({\mathbb{B}}\)-valued model of set theory \(V^{({\mathbb{B}})}\); in this paper a category-theoretic translation of \(V^{({\mathbb{B}})}\) is given, in the form of a \({\mathbb{B}}\)-valued model of category theory. The usual category-theoretic translation of \(V^{({\mathbb{B}})}\), namely the category of sheaves Shv(\({\mathbb{B}})\), appears as an image of the \({\mathbb{B}}\)- valued model. The \({\mathbb{B}}\)-valued model lives in a category MOD(\({\mathbb{B}})\), which is intended to be the category of all \({\mathbb{B}}\)- valued models. The last part of the paper investigates Easton's construction, which is the construction of \(V^{({\mathbb{B}})}\) for a `large' \({\mathbb{B}}\). The construction (in MOD(\({\mathbb{B}}))\) of the \({\mathbb{B}}\)- valued model of category theory can still be carried out in this case, though the construction of Shv(\({\mathbb{B}})\) fails.
Mathematics Subject Classification ID: 03E40
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 4016897
zbMATH Keywords: Easton forcing
zbMATH Keywords: Boolean-valued model of category theory
zbMATH Keywords: complete Boolean algebra
zbMATH Keywords: category of sheaves
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90010-1
OpenAlex ID: W2060395830
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description: scientific article; zbMATH DE number 4016897