High Quality Content by WIKIPEDIA articles! Freiling's axiom of symmetry (AX) is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidson but the mathematics behind it goes back to Wac?aw Sierpi?ski. Let A be the set of functions mapping numbers in the unit interval [0,1] to countable subsets of the same interval. The axiom AX states:For every f in A, there exist x and y such that x is not in f(y) and y is not in f(x). A theorem of Sierpi?ski says that under the assumptions of ZFC set theory, AX is equivalent to the negation of the continuum hypothesis (CH). Although Sierpi?ski did not formally promote this as evidence against CH, it is likely that he understood the paradoxical implications in a similar manner as Freiling. Sierpi?ski's theorem answered a question of Hugo Steinhaus and was proved long before the independence of CH had been established by Kurt Godel and Paul Cohen. It was Stewart Davidson who first suggested to Freiling that Sierpi?ski's theorem should be considered as evidence against CH.

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