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 \({2+2+2+\ldots = ?}\)”.
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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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Tachtsis, E. On the existence of permutations of infinite sets without fixed points in set theory without choice. Acta Math. Hungar. 157, 281–300 (2019). https://doi.org/10.1007/s10474-018-0869-9
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DOI: https://doi.org/10.1007/s10474-018-0869-9
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