Freiling's axiom of symmetry

Freiling's axiom of symmetry (${\displaystyle {\texttt {AX}}}$) 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 ${\displaystyle A\subseteq {\mathcal {P}}([0,1])^{[0,1]}}$ denote the set of all functions from ${\displaystyle [0,1]}$ to countable subsets of ${\displaystyle [0,1]}$. (In other words, ${\displaystyle A=\left({\big [}[0,1]{\big ]}^{\leq \omega }\right)^{[0,1]}}$ where ${\displaystyle [X]^{\leq \kappa }}$ is the collection of subsets of ${\displaystyle X}$ of cardinality at most ${\displaystyle \kappa }$.)

The axiom ${\displaystyle {\texttt {AX}}}$ then states:

For every ${\displaystyle f\in A}$, there exist ${\displaystyle x,y\in [0,1]}$ such that ${\displaystyle x\not \in f(y)}$ and ${\displaystyle y\not \in f(x)}$.

A theorem of Sierpiński says that under the assumptions of ZFC set theory, ${\displaystyle {\texttt {AX}}}$ 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.