Freiling's axiom of symmetry is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidsonbut the mathematics behind it goes back to Wacław Sierpiński. Let denote the set of all functions from to countable subsets of . The axiom states: For every , there exist such that and . Freiling claims that probabilistic intuition strongly supports this proposition while others disagree. There are several versions of the axiom, some of which are discussed below.