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


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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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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