Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T14:04:16.024Z Has data issue: false hasContentIssue false
  • English
  • Français

Unions of well-ordered sets

Part of: Set theory

Published online by Cambridge University Press:  09 April 2009

Paul Howard
Paul Howard
Affiliation:
Eastern Michigan University, Ypsilanti, MI 48197, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In Zermelo-Fraenkel set theory weakened to permit the existence of atoms and without the axiom of choice we investigate the deductive strength of five statements which make assertions about the cardinality of the union of a well-ordered collection of sets. All five of the statements considered are consequences of the axiom of choice.

Keywords

axiom of choicecountable union theoremwell-ordered set.

MSC classification

Secondary: 03E35: Consistency and independence results 03E25: Axiom of choice and related propositions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 56 , Issue 1 , February 1994 , pp. 117 - 124
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]N. Brunner and P. Howard, ‘Russell's alternative to the axiom of choice’, Math. Logik Grundlag. Math., to appear.Google Scholar
[2]Cantor, G., ‘Mitteilungen zur Lehre vom Transfiniten’, Zeitschrift f. Philosophie und philosophische Kritik 91 (1887), 81–125.Google Scholar
[3]P. Howard, ‘The countable union theorem does not imply the axiom of choice for countable collections of countable sets’, Notre Dame J. Formal Logic, to appear.Google Scholar
[4]Kunen, K., Set theory, an introduction to independence proofs, Stud. in Logic Found. Math. 102 (North-Holland, Amsterdam, 1980).Google Scholar
[5]Moore, G., Zermelo's axiom of choice, Stud. Hist. Math. Phys. Sci., 8 (Springer, New York, 1982).CrossRefGoogle Scholar
[6]Rubin, H. and Rubin, J., Equivalents of the axiom of choice, II, Stud. in Logic Found. Math. 116 (North-Holland, Amsterdam, 1985).Google Scholar
[7]Sierpinski, W., ‘L'axiome de M. Zermelo et son rôle dans la théorie des ensembles et l'analyse’, Bull. Acad. Sci. Cracovie, Class des Sciences Math., Série A (1918), 97–152.Google Scholar