Well-ordering and choice in toposes
type: Item
instance of: scholarly article
title: Well-ordering and choice in toposes
zbMATH Open document ID: 0645.18001
DOI: 10.1016/0022-4049(88)90113-2
author: Mbila-Mambu Mawanda
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: This paper gives a new illustration of an interesting feature of topos theory: results or concepts equivalent in the context of sets are no more equivalent in the general context of toposes. It is a well-known result by Diaconescu that a topos verifying the axiom of choice (every epi splits) is boolean. What about Zermelo theorem on the equivalence between the axiom of choice and the fact that every set can be well-ordered? The answer is as follows: the author introduces a new definition of well-ordering, which coincides with the usual one in sets, and constructs non-boolean toposes such that every object can be well ordered. In section 2, a new definition for the axiom of choice in a topos is given, which is equivalent to the former one for sets and which does not imply booleanness as it is proved by the construction of examples.
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 03E25
zbMATH DE Number: 4051857
zbMATH Keywords: Zermelo theorem
zbMATH Keywords: well-ordering
zbMATH Keywords: non-boolean toposes
zbMATH Keywords: axiom of choice in a topos
reviewed by: Marie-Françoise Roy
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90113-2
OpenAlex ID: W2030860303
cites work: Finiteness: Another aspect of topoi
cites work: Axiom of Choice and Complementation
cites work: Q4145861
cites work: Q3889051
description: scientific article; zbMATH DE number 4051857