Freiling's axiom of symmetry (\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 denote the set of all functions from [0,1] to countable subsets of [0,1]. (In other words, .) The axiom \texttt{AX} states: A theorem of Sierpiński says that under the assumptions of ZFC set theory, \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.

Freiling's argument

Fix a function f in A. We will consider a thought experiment that involves throwing two darts at the unit interval. We are not 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 \texttt{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).

Relation to the (Generalised) Continuum Hypothesis

Fix \kappa, an infinite cardinal (e.g. ). Let be the statement: there is no map from sets to sets of size \leq\kappa for which either x\in f(y), or y\in f(x),. Claim:. Proof: Part I : Suppose. Then there exists a bijection. Setting defined via, it is easy to see that this demonstrates the failure of Freiling's axiom. Part II : Suppose that Freiling's axiom fails. Then fix some f, to verify this fact. Define an order relation on by A\leq_{f} B iff A\in f(B). This relation is total and every point has \leq\kappa many predecessors. Define now a strictly increasing chain as follows: at each stage choose. This process can be carried out since for every ordinal, is a union of many sets of size ; thus is of size and so is a strict subset of. We also have that this sequence is cofinal in the order defined, i.e. every member of is \leq_{f}, some. (For otherwise if is not \leq_{f}, some A_{\alpha}, then since the order is total ; implying B, has many predecessors; a contradiction.) Thus we may well-define a map by. So which is union of many sets each of size. Hence. Note that so we can easily rearrange things to obtain that the above-mentioned form of Freiling's axiom. The above can be made more precise:. This shows (together with the fact that the continuum hypothesis is independent of choice) a precise way in which the (generalised) continuum hypothesis is an extension of the axiom of choice.

Objections to Freiling's argument

Freiling's argument is not widely accepted because of the following two problems with it (which Freiling was well aware of and discussed in his paper).

Connection to graph theory

Using the fact that in ZFC, we have (see above), it is not hard to see that the failure of the axiom of symmetry — and thus the success of — is equivalent to the following combinatorial principle for graphs: In the case of, this translates to: Thus in the context of ZFC, the failure of a Freiling axiom is equivalent to the existence of a specific kind of choice function.