Freiling's axiom of symmetry (
) 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.
then states: A theorem of Sierpiński says that under the assumptions of ZFC set theory,
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.
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.
Freiling now makes two generalizations: The axiom
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).
to build an ascending chain in
Given this function, it is straightforward to see that this demonstrates the failure of Freiling's axiom.
(sometimes called the pushforward of the standard ordering on
): Suppose that Freiling's axiom fails.
Define an order relation on
This relation is total and every point has
Define now a strictly increasing chain
We also have that this sequence is cofinal in the order defined, i.e. every member of
so we can easily rearrange things to obtain that
the above-mentioned form of Freiling's axiom.
{\displaystyle {\texttt {ZF}}\vdash ({\texttt {AC}}_{{\mathcal {P}}(\kappa )}+\neg {\texttt {AX}}_{\kappa })\leftrightarrow {\texttt {CH}}_{\kappa }\,}
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.
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).
(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.