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 real numbers in the unit interval [0,1] to countable subsets of the same interval. The axiom AX states:

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.

Fix a function f in A. We will consider a thought experiment that involves throwing two darts at the unit interval. We aren't able to physically determine with infinite accuracy the actual values of the numbers x and y that are hit. Likewise, the question of whether "y is in f(x)" cannot actually be physically computed. Nevertheless, if f really is a function, then this question is a meaningful one and will have a definite "yes" or "no" answer.

Now wait until after the first dart, x, is thrown and then assess the chances that the second dart y will be in f(x). Since x is now fixed, f(x) is a fixed countable set and has Lebesgue measure zero. Therefore, this event, with x fixed, has probability zero. Freiling now makes two generalizations:

The axiom AX is now justified based on the principle that what will predictably happen every time this experiment is performed, should at the very least be possible. Hence there should exist two real numbers x, y such that x is not in f(y) and y is not in f(x).

Fix ${\displaystyle \kappa \,}$ an infinite cardinal (e.g. ${\displaystyle \aleph _{0}\,}$). Let ${\displaystyle {\texttt {AX}}_{\kappa }.\,}$ be the statement: there is no map ${\displaystyle f:{\mathcal {P}}(\kappa )\to {\mathcal {P}}{\mathcal {P}}(\kappa )\,}$ from sets to sets of size ${\displaystyle \leq \kappa }$ for which ${\displaystyle (\forall {x,y\in {\mathcal {P}}(\kappa )})\,}$ either ${\displaystyle x\in f(y)\,}$ or ${\displaystyle y\in f(x)\,}$.

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