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FACTORIALS OF INFINITE CARDINALS IN ZF PART II: CONSISTENCY RESULTS

Published online by Cambridge University Press:  04 November 2019

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:shen_guozhen@outlook.com
JIACHEN YUAN
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES AND LPMC NANKAI UNIVERSITY TIANJIN300071PEOPLE’S REPUBLIC OF CHINAE-mail:819081@nankai.edu.cn

Abstract

For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF:

(1) There is an infinite set x such that $|\wp \left( x \right)| < |{\cal S}\left( x \right)| < |se{q^{1 - 1}}\left( x \right)| < |seq\left( x \right)|$, where $\wp \left( x \right)$ is the power set of x, seq (x) is the set of all finite sequences of elements of x, and seq1-1 (x) is the set of all finite sequences of elements of x without repetition.

(2) There is a Dedekind infinite set x such that $|{\cal S}\left( x \right)| < |{[x]^3}|$ and such that there exists a surjection from x onto ${\cal S}\left( x \right)$.

(3) There is an infinite set x such that there is a finite-to-one function from ${\cal S}\left( x \right)$ into x.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2019 

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