The axiom of choice for countable collections of countable sets does not imply the countable union theorem
type: Item
instance of: scholarly article
title: The axiom of choice for countable collections of countable sets does not imply the countable union theorem
zbMATH Open document ID: 0760.03014
author: Paul Howard
published in: Notre Dame Journal of Formal Logic
publication date: 1993-02-22T00:00:00Z
review text: As has been shown by \textit{U. Felgner} [Comment. Math. Helv. 49, 114-124 (1974; Zbl 0283.02046)], in ZF the axiom of choice for families of nonempty well-orderable sets does not imply the countable union theorem. In Felgner's model \(\aleph_ 1\) is singular and the question arises if a similar independence result can be obtained by means of the Fraenkel- Mostowski method. In the paper under review the following theorem is proved by this method: The countable union theorem does not follow from the axiom of choice for well-ordered families of nonempty sets, each of cardinality not greater than or equal to \(2^{\aleph_ 0}\).
reviewed by: Norbert Brunner
Mathematics Subject Classification ID: 03E25
zbMATH DE Number: 120389
zbMATH Keywords: countable union theorem
zbMATH Keywords: Fraenkel-Mostowski method
zbMATH Keywords: axiom of choice for well-ordered families of nonempty sets
Wikidata QID: Q114598311
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093636102
OpenAlex ID: W2055381384
DOI: 10.1305/NDJFL/1093636102
description: scientific article; zbMATH DE number 120389