## Notre Dame Journal of Formal Logic

### A Note on Weakly Dedekind Finite Sets

#### Abstract

A set $A$ is Dedekind infinite if there is a one-to-one function from $\omega$ into $A$. A set $A$ is weakly Dedekind infinite if there is a function from $A$ onto $\omega$; otherwise $A$ is weakly Dedekind finite. For a set $M$, let $\operatorname{dfin}^{*}(M)$ denote the set of all weakly Dedekind finite subsets of $M$. In this paper, we prove, in Zermelo–Fraenkel (ZF) set theory, that $|\operatorname{dfin}^{*}(M)|\lt |\mathcal{P}(M)|$ if $\operatorname{dfin}^{*}(M)$ is Dedekind infinite, whereas $|\operatorname{dfin}^{*}(M)|\lt |\mathcal{P}(M)|$ 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

https://projecteuclid.org/euclid.ndjfl/1406034055

Digital Object Identifier
doi:10.1215/00294527-2688096

Mathematical Reviews number (MathSciNet)
MR3263536

Zentralblatt MATH identifier
1338.03097

#### 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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