Freiling's axiom of symmetry

From Free net encyclopedia

Freiling's axiom of symmetry (AX) is a set-theoretic axiom proposed by Chris Freiling. The conjunction of AX with the axiom of choice entails that the continuum hypothesis does not hold.

Let A be the set of functions mapping real numbers to countable sets of real numbers. Given a function f in A, and some arbitrary real numbers x and y, it is generally held that x is in f(y) with probability 0, i.e. x is not in f(y) with probability 1. Similarly, y is not in f(x) with probability 1. AX states:

For every f in A, there exist x and y such that x is not in f(y) and y is not in f(x).

Freiling claims that probabilistic intuition strongly supports this proposition.

Freiling's argument (informal version)

An informal version of Freiling's argument against the continuum hypothesis goes as follows. Suppose the continuum hypothesis holds (and also assume the axiom of choice). Then we can well order the reals so that for each real there are only a countable number of smaller reals. (This ordering has nothing to do with the usual order on the reals.) Now throw a dart at the unit square to produce a "random" pair (x,y) of real numbers. The chance that x is smaller than y (in this well ordering) is zero because given y there are only a countable number of x smaller than y and an uncountable number larger than y. Similarly the chance that y is smaller than x is also zero. This gives a contradiction, as the chances must add up to 1.

Objections to Freiling's argument

Freiling's argument is not widely accepted because of the following two problems with it.

• Opponents argue that probabilistic intuition often tacitly assumes that all sets and functions under consideration are measurable, and hence should not be used together with the axiom of choice, since an invocation of the axiom of choice typically generates non-measurable sets. The naive probabilistic intuition used is well known to fail badly for non-measurable sets; see Banach–Tarski paradox for the most blatant example.
• A minor variation of his argument gives a contradiction with the axiom of choice whether or not one accepts the continuum hypothesis. One just replaces "countable" with "cardinality less than the continuum" in his argument. So his argument seems to be more an argument against the possibility of well ordering the reals than against the continuum hypothesis.

References

• Chris Freiling, Axioms of symmetry: throwing darts at the real number line. J. Symbolic Logic 51 (1986), no. 1, 190-200.