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FACTORIALS OF INFINITE CARDINALS IN ZF PART I: ZF RESULTS

Published online by Cambridge University Press:  04 November 2019

GUOZHEN SHEN  and
JIACHEN YUAN
GUOZHEN SHEN
Affiliation:
INSTITUTE OF MATHEMATICS ACADEMY OF MATHEMATICS AND SYSTEMS SCIENCE CHINESE ACADEMY OF SCIENCES BEIJING100190PEOPLE’S REPUBLIC OF CHINA and SCHOOL OF MATHEMATICAL SCIENCES UNIVERSITY OF CHINESE ACADEMY OF SCIENCES BEIJING 100049 PEOPLE’S REPUBLIC OF CHINAE-mail:[email protected]
JIACHEN YUAN
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES AND LPMC NANKAI UNIVERSITY TIANJIN300071PEOPLE’S REPUBLIC OF CHINAE-mail:[email protected]
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Abstract

For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove in ZF (without the axiom of choice) several results concerning this notion, among which are the following:

(1) For all sets x such that ${\cal S}\left( x \right)$ is Dedekind infinite, $\left| {{{\cal S}_{{\rm{fin}}}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ and there are no finite-to-one functions from ${\cal S}\left( x \right)$ into ${{\cal S}_{{\rm{fin}}}}\left( x \right)$, where ${{\cal S}_{{\rm{fin}}}}\left( x \right)$ denotes the set of all permutations of x which move only finitely many elements.

(2) For all sets x such that ${\cal S}\left( x \right)$ is Dedekind infinite, $\left| {{\rm{seq}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ and there are no finite-to-one functions from ${\cal S}\left( x \right)$ into seq (x), where seq (x) denotes the set of all finite sequences of elements of x.

(3) For all infinite sets x such that there exists a permutation of x without fixed points, there are no finite-to-one functions from ${\cal S}\left( x \right)$ into x.

(4) For all sets x, $|{[x]^2}| < \left| {{\cal S}\left( x \right)} \right|$.

Keywords

Primary 03E1003E25ZFcardinalfactorialpermutationfinite-to-one functionDedekind finite
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 224 - 243
Copyright
Copyright © The Association for Symbolic Logic 2019 

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