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 Davidsonbut the mathematics behind it goes back to Wacław Sierpiński.

Let "A" be the set of functions mapping numbers in the unit interval [0,1] to countable subsets of the same interval. The axiom "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").

A theorem of Sierpiński says that under the assumptions of ZFC set theory, "AX" is equivalent to the negation of the continuum hypothesis (CH). Although Sierpiński did not formally promote this as evidence against CH, itis likely that he understood the paradoxical implications in a similar manneras Freiling. Sierpiński's theorem answered a question of Hugo Steinhausand was proved long before the independence of CH had been established by
Kurt Gödel and Paul Cohen. It was Stewart Davidson who first suggestedto Freiling that Sierpiński's theorem should be considered as evidence againstCH.

Freiling claims that probabilistic intuition strongly supports this propositionwhile others disagree. There are several versions of the axiom, some of whichare 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 probably 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:

* Since we can predict with virtual certainty that "y" is not in "f"("x")" after the first dart is thrown, and since this prediction is valid no matter what the first dart does, we should be able to make this prediction before the first dart is thrown. This is not to say that we still have a measurable event, rather it is an intuition about the nature of being predictable.

* Since "y" is not in "f"("x")" is predictably true, by the symmetry of the order in which the darts were thrown (hence the name "axiom of symmetry") we should also be able to predict with virtual certainty that "x" is not in "f"("y")".

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").

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 defines a cardinal κ as the smallest possible cardinality of a non-null set; replacing "countable" with "cardinality less than κ" in Freiling's argument (and modifying the definition of the set "A" accordingly) now establishes an outright contradiction. Fact|date=February 2007So his argument seems to be more an argument against the possibility of well ordering the reals than against the continuum hypothesis.

Connection with Solovay's forcing

Freiling's argument appeals to the intuitive idea of throwing darts at the real line. In order to consistently throw darts at the real line, every subset of the real line must be measurable. The reason is that given a set S, if you throw a dart, it becomes a "fixed" and "definite" real number, and you can check if it is in the set S or not. Throw the dart again and again, and "define" the measure of any set as the limit of the ratio: (# that land in S) / (# thrown).

Since every throw is statistically independent, this converges to a unique value. The value defines a countably additive, translation invariant measure which coincides with Lebesgue measure for intervals. This is why Freiling's argument is really about the axiom of choice, not so much about the continuum hypothesis.

The existence of a model of set theory in which the reals support a translation-invariant measure is consistent assuming that it is consistent to have an inaccessible cardinal, a set which can serve as a model for ZFC. This was proven by Robert Solovay who, following earlier work of Paul Cohen, introduced the notion of a random real number. The definition of measure of a set as a ratio is closely related to Solovay's definition of measure.

Counter-arguments

*Freiling's argument never assumes, tacitly or otherwise, that all sets and functions under consideration are measurable. Rather it applies these traditional notions in the non-traditional context of time. After the first dart is thrown, it is no longer a random variable, but is a fixed and definite real number. After it is thrown the target set for the second dart is indeed a fixed Lebesgue measurable set with measure zero. The rest of the argument has nothing to do with traditional mathematical concepts, but relies on Freiling's intuition involving the nature of predictability and the symmetry of time. "The real number line can't tell which dart was thrown first and which was second".

*Although Lebesgue-measurable events are generally assumed to have a well defined probability, there is nothing to suggest that to have a well defined probability an event must be measurable. This is nothing new; extensions of Lebesgue measure have long been considered by mathematicians.

*It is undeniable that extensions of this intuition will contradict the axiom of choice, but they are extensions. The axiom itself is compatible with the axiom of choice. Freiling's argument is that with ZFC+CH one is forced to accept even stronger contradictions to natural intuition than one would have to accept with ZFC alone.

Extensions and alternate versions

There are several variations to consider. Suppose first that "A" was theset of functions mapping [0,1] to finite subsets of [0,1] . Suppose that we madethe corresponding assertion:

: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").

This statement is not just intuitively true, it can be proved.However, let's now make a minor adjustment to accommodate three darts. Let "A" be the set of functions that map pairs of numbers from [0,1] tofinite subsets of [0,1] . After two darts "x", "y" are thrown, we canpredict that the third dart, "z", will not be in the set assigned to the pair {"x","y"}. As before, since this prediction is always the same, it shouldbe equally valid before the first two darts are thrown. Then by symmetry, we also expect that "x" is not in the set assigned to {"y","z"} and that "y" is not in the set assigned to {"x","z"}. Hence we should at leastbelieve that:

:For every "f" in "A" there exist "x", "y", and "z" such that "z" is not in "f"({"x","y"}), "y" is not in "f"({"x","z"}), and "x" is not in "f"({"y","z"}).

Like the original version given above, this is equivalent in ZFC to thenegation of the continuum hypothesis. The advantage of this version is thatit requires only intuition about "finite" sets having probability zero.

One can throw more and more darts this way, and with each successive dart, thelower bound on the continuum grows. If one allows versions for infinitely manydarts then the continuum is bounded below by a Jónsson cardinal, which is a typeof large cardinal. But if one tries this with ω+1 darts then anothercontradiction is reached with the Axiom of Choice.

References

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

*David Mumford, "The dawning of the age of stochasticity", in "Mathematics: Frontiers and Perspectives 2000", American Mathematical Society, 1999, 197-218.

*Wacław Sierpiński, "Hypothèse du Continu", Chelsea Publishing Co., New York, 1956.

*John Simms, "Traditional Cavalieri principles applied to the modern notion of area", "J. Philosophical Logic" 18 (1989), 275-314.