Freiling's Axiom of Symmetry

Freiling's Axiom Of Symmetry

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 Wacław Sierpiński.

Let A be the set of functions mapping numbers in the unit interval 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 Sierpiński says that under the assumptions of ZFC set theory, AX is equivalent to the negation of the continuum hypothesis (CH). Sierpiński's theorem answered a question of Hugo Steinhaus and was proved long before the independence of CH had been established by Kurt Gödel and Paul Cohen.

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.