Freiling's Axiom of Symmetry - Objections To Freiling's Argument

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).

• The naive probabilistic intuition used by Freiling tacitly assumes that there is a well-behaved way to associate a probability to any subset of the reals. But the mathematical formalization of the notion of "probability" uses the notion of measure, yet the axiom of choice implies the existence of non-measurable subsets, even of the unit interval. Some examples of this are the Banach–Tarski paradox and the existence of Vitali sets.
• A minor variation of his argument gives a contradiction with the axiom of choice whether or not one accepts the continuum hypothesis, if one replaces countable additivity of probability by additivity for cardinals less than the continuum. (Freiling used a similar argument to claim that Martin's axiom is false.) It is not clear why Freiling's intuition should be any less applicable in this instance, if it applies at all. (Maddy 1988, p. 500) So Freiling's argument seems to be more an argument against the possibility of well ordering the reals than against the continuum hypothesis.