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. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the axiom of choice is an axiom of set theory. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

Let A be the set of functions mapping real numbers to 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).

Freiling claims that probabilistic intuition strongly supports this proposition. Opponents argue that probabilistic intuition often tacitly assumes that all sets and functions under consideration are measurable, and hence should not be used together with the axiom of choice, since an invocation of the axiom of choice typically generates non-measurable sets. (See Banach-Tarski paradox as the most blatant example.) In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In mathematics, the axiom of choice is an axiom of set theory. ... The Banach-Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. ...