Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T14:26:12.224Z Has data issue: false hasContentIssue false
  • English
  • Français

FACTORIALS OF INFINITE CARDINALS IN ZF PART II: CONSISTENCY 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: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
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Primary 03E1003E25ZFcardinalfactorialpermutationpermutation modelfinite-to-one function
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 244 - 270
Copyright
Copyright © The Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Birkhoff, G., Lattice Theory, third ed., American Mathematical Society Colloquium Publications, vol. 25, American Mathematical Society, Providence, R. I., 1967.Google Scholar
Dawson, J. W. Jr. and Howard, P. E., Factorials of infinite cardinals. Fundamenta Mathematicae, vol. 93 (1976), pp. 185–195.Google Scholar
Forster, T., Finite-to-one maps, this Journal, vol. 68 (2003), pp. 1251–1253.Google Scholar
Grätzer, G., General Lattice Theory, second ed., Birkhäuser, Basel, 1998.Google Scholar
Halbeisen, L., Combinatorial Set Theory: With a Gentle Introduction to Forcing, second ed., Springer Monographs in Mathematics, Springer, Cham, 2017.CrossRefGoogle Scholar
Halbeisen, L., A weird relation between two cardinals. Archive for Mathematical Logic, vol. 57 (2018), pp. 593–599.CrossRefGoogle Scholar
Halbeisen, L. and Shelah, S., Consequences of arithmetic for set theory, this Journal, vol. 59 (1994), pp. 30–40.Google Scholar
Halbeisen, L. and Shelah, S., Relations between some cardinals in the absence of the axiom of choice. Bulletin of Symbolic Logic, vol. 7 (2001), pp. 237–261.CrossRefGoogle Scholar
Howard, P. and Rubin, J. E., Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, R. I., 1998.Google Scholar
Jech, T., The Axiom of Choice, Studies in Logic and the Foundations of Mathematics, vol. 75, North-Holland, Amsterdam, 1973.Google Scholar
Rubin, J. E., Non-constructive properties of cardinal numbers. Israel Journal of Mathematics, vol. 10 (1971), pp. 504–525.CrossRefGoogle Scholar
Shen, G., Generalizations of Cantor’s theorem in ZF. Mathematical Logic Quarterly, vol. 63 (2017), pp. 428–436.Google Scholar
Shen, G., A note on strongly almost disjoint families. Notre Dame Journal of Formal Logic, accepted, 2019.Google Scholar
Shen, G. and Yuan, J., Factorials of infinite cardinals in ZF, Part I: ZF results, submitted, 2019.Google Scholar
Sonpanow, N. and Vejjajiva, P., Factorials and the finite sequences of sets. Mathematical Logic Quarterly, vol. 65 (2019), pp. 116–120.CrossRefGoogle Scholar
Specker, E., Verallgemeinerte Kontinuumshypothese und Auswahlaxiom. Archiv der Mathematik, vol. 5 (1954), pp. 332–337.Google Scholar
Tachtsis, E., On the existence of permutations of infinite sets without fixed points in set theory without choice. Acta Mathematica Hungarica, vol. 157 (2019), pp. 281–300.CrossRefGoogle Scholar
Truss, J., Dualisation of a result of Specker’s. Journal of the London Mathematical Society, vol. 6 (1973), pp. 286–288.CrossRefGoogle Scholar
Vejjajiva, P. and Panasawatwong, S., A note on weakly Dedekind finite sets. Notre Dame Journal of Formal Logic, vol. 55 (2014), pp. 413–417.CrossRefGoogle Scholar