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

摘要

In set theory without the Axiom of Choice (AC), we investigate the problem of the deductive strength of the statement For every infinite set X, there exists a permutation ofXwithout fixed points (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 thatf2=idX (ISAE). Among several results, we prove that EPWFP is strictly weaker than ISAE in ZFA set theory. We also settle a plethora of open problems on the relative strength of 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+...=?.