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Assuming axiom of choice , for any set $S$ with more than one point , there exist a bijection $f:S \to S$ such that $f(s) \ne s , \forall s \in S$ . Is the converse true , i.e. Does the statement " For every set $S$ with more than one point , there exist a bijection $f:S \to S$ such that $f(s) \ne s , \forall s \in S$ " implies Axiom of Choice ? Or at least can we prove that any set with more than point has a permutation with no fixed point , without axiom of choice ? Please help. Thanks in advance

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marked as duplicate by Asaf Karagila axiom-of-choice Apr 13 '15 at 15:25

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  • $\begingroup$ My answer in the duplicate shows that it is much weaker than choice. It is not provable without choice, and there are threads here with such example, but I have to go and it will have to wait (or you can search for it yourself). $\endgroup$ – Asaf Karagila Apr 13 '15 at 15:26
  • $\begingroup$ @AsafKaragila : I do not see any reference there of why "it is not provable without choice " $\endgroup$ – user228168 Apr 13 '15 at 15:29
  • $\begingroup$ As I said, this is in a different question, and you can look for it yourself or wait until I am by a keyboard to do that for you. $\endgroup$ – Asaf Karagila Apr 13 '15 at 15:33
  • $\begingroup$ math.stackexchange.com/questions/103161/… $\endgroup$ – Asaf Karagila Apr 13 '15 at 15:36
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This statement allready follows from the addition theorem, which states that $A \times \{0,1\}$ is equipotent with $A$ for every infinite $A$ (as on $A \times \{0,1\}$, $(a,x) \mapsto (a, 1+x)$ is fixed-point-free). The addition theorem is known to be strictly weaker than the axiom of choice (a result by Sageev).

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  • $\begingroup$ True it follows , but I didn't know that this is strictly weaker than axiom of choice , could you please give a reference ? $\endgroup$ – user228168 Apr 13 '15 at 15:26
  • $\begingroup$ Sageev, G.: Notices Amer. Math. Soc. 20 (1973), A22. $\endgroup$ – martini Apr 13 '15 at 15:28
  • $\begingroup$ The paper was fully published, in 180 painful pages of forcing in pre-Shoenfield times. $\endgroup$ – Asaf Karagila Apr 13 '15 at 15:34