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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: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
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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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References

REFERENCES

Blass, A., Power-Dedekind finiteness, manuscript, 2013.Google Scholar
Dawson, J. W. Jr. and Howard, P. E., Factorials of infinite cardinals. Fundamenta Mathematicae, vol. 93 (1976), pp. 185–195.CrossRefGoogle Scholar
Degen, J. W., Some aspects and examples of infinity notions. Mathematical Logic Quarterly, vol. 40 (1994), pp. 111–124.CrossRefGoogle Scholar
Ellentuck, E., Generalized idempotence in cardinal arithmetic. Fundamenta Mathematicae, vol. 58 (1966), pp. 241–258.CrossRefGoogle Scholar
Forster, T., Finite-to-one maps, this Journal, vol. 68 (2003), pp. 1251–1253.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. and Shelah, S., Consequences of arithmetic for set theory, this Journal, vol. 59 (1994), pp. 30–40.Google Scholar
Herrlich, L., The finite and the infinite. Applied Categorical Structures, vol. 19 (2011), pp. 455–468.CrossRefGoogle Scholar
Levy, A., The independence of various definitions of finiteness. Fundamenta Mathematicae, vol. 46 (1958), pp. 1–13.CrossRefGoogle Scholar
Levy, A., Basic Set Theory, Perspectives in Mathematical Logic, Springer, Berlin, 1979.CrossRefGoogle Scholar
Shen, G., Generalizations of Cantor’s theorem in ZF. Mathematical Logic Quarterly, vol. 63 (2017), pp. 428–436.Google Scholar
Sonpanow, N. and Vejjajiva, P., A finite-to-one map from the permutations on a set. Bulletin of the Australian Mathematical Society, vol. 95 (2017), pp. 177–182.CrossRefGoogle Scholar
Sonpanow, N. and Vejjajiva, P., Some properties of infinite factorials. Mathematical Logic Quarterly, vol. 64 (2018), pp. 201–206.CrossRefGoogle 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.CrossRefGoogle 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
Tarski, A., Sur les ensembles finis. Fundamenta Mathematicae, vol. 6 (1924), pp. 45–95.CrossRefGoogle Scholar
Truss, J., Classes of Dedekind finite cardinals. Fundamenta Mathematicae, vol. 84 (1974), pp. 187–208.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