Freiling's axiom of symmetry

Freiling's Axiom of Symmetry (AX) is a set-theoretic axiom proposed by Chris Freiling. The conjunction of AX with the axiom of choice entails that the continuum hypothesis does not hold.

Let A be the set of functions mapping real numbers into countable sets of real numbers. Given a function f in A, and some arbitrary real numbers x and y, it is generally held that x is in f(y) with probability 0, i.e. x is not in f(y) with probability 1. Similarly, y is not in f(x) with probability 1. 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).

Probabilistic intuition strongly supports this proposition.