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Acta Mathematica Hungarica

, Volume 157, Issue 2, pp 281–300 | Cite as

On the existence of permutations of infinite sets without fixed points in set theory without choice

  • E. TachtsisEmail author
Article
  • 43 Downloads

Abstract

In set theory without the Axiom of Choice (\({\mathsf{AC}}\)), we investigate the problem of the deductive strength of the statement “For every infinite set X, there exists a permutation of\({X}\)without fixed points” (\({\mathsf{EPWFP}}\)) as well as of the formally stronger statement “For every infinite set X, there exists a permutation f of X without fixed points and such that\({f^{2}=\mathrm{id}_{X}}\)” (\({\mathsf{ISAE}}\)). Among several results, we prove that \({\mathsf{EPWFP}}\) is strictly weaker than \({\mathsf{ISAE}}\) in \({\mathsf{ZFA}}\) set theory.

We also settle a plethora of open problems on the relative strength of \({\mathsf{ISAE}}\) which are left open in Sonpanow and Vejjajiva “A finite-to-one map from the permutations on a set”, and in Herrlich and Tachtsis “On the number of Russell’s socks or \({2+2+2+\ldots = ?}\)”.

Mathematics Subject Classification

primary 03E25 secondary 03E35 05A05 05A18 

Key words and phrases

axiom of choice weak axiom of choice permutation of a set fixed point of a permutation almost even set Fraenkel–Mostowski model of \({\mathsf{ZFA}}\) symmetric model of \({\mathsf{ZF}}\) Jech–Sochor First Embedding Theorem Pincus’ transfer theorem 

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Notes

Acknowledgement

We are most thankful to the anonymous referee for careful review work, whose suggestions also removed an error from the original version, and improved the quality of our paper.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of the AegeanKarlovassiGreece

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