Axiom proposed by Chris Freiling, referred to as AX. If AX is accepted then 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 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).