Notre Dame Journal of Formal Logic

A Note on Weakly Dedekind Finite Sets

Pimpen Vejjajiva and Supakun Panasawatwong

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Abstract

A set A is Dedekind infinite if there is a one-to-one function from ω into A. A set A is weakly Dedekind infinite if there is a function from A onto ω; otherwise A is weakly Dedekind finite. For a set M, let dfin(M) denote the set of all weakly Dedekind finite subsets of M. In this paper, we prove, in Zermelo–Fraenkel (ZF) set theory, that |dfin(M)|<|P(M)| if dfin(M) is Dedekind infinite, whereas |dfin(M)|<|P(M)| cannot be proved from ZF for an arbitrary M.

Article information

Source
Notre Dame J. Formal Logic, Volume 55, Number 3 (2014), 413-417.

Dates
First available in Project Euclid: 22 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1406034055

Digital Object Identifier
doi:10.1215/00294527-2688096

Mathematical Reviews number (MathSciNet)
MR3263536

Zentralblatt MATH identifier
1338.03097

Subjects
Primary: 03E10: Ordinal and cardinal numbers 03E25: Axiom of choice and related propositions

Keywords
weakly Dedekind finite Dedekind infinite axiom of choice

Citation

Vejjajiva, Pimpen; Panasawatwong, Supakun. A Note on Weakly Dedekind Finite Sets. Notre Dame J. Formal Logic 55 (2014), no. 3, 413--417. doi:10.1215/00294527-2688096. https://projecteuclid.org/euclid.ndjfl/1406034055


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