A New Perspective on the Continuum Hypothesis:  Why Freiling’s Axiom of Symmetry is Irrelevant
by William C. Abram

Abrstract:  Since Georg Cantor proposed the Continuum Hypothesis in 1877, there has been much debate about the plausibility of the Continuum Hypothesis. With the results of Kurt Gödel in 1940 and Paul Cohen in 1963, Hilbert’s 1st problem was proved independent of the axioms of Zermelo-Fraenkel Set Theory. Therefore, the proof or negation of the Continuum Hypothesis, if proof or negation is desired, must be supplied by an additional axiom. Such an axiom must be intuitively true and consistent with the axioms of Zermelo-Fraenkel Set Theory. Herein I contend that Freiling’s Axiom of Symmetry, which is equivalent to the negation of the Continuum Hypothesis, is not the desired axiom.

Naïve and Zermelo-Fraenkel Set Theories

The Naïve definition of sets as well-defined collections of objects rapidly proves inconsistent under formal treatment. As Bertrand Russel discovered in 1901, for the set x = {y: y is not in y}, x is in x iff x is not in x. This result, known as Russel’s Paradox, was a motivating force for the development of axiomatic set theories.

Today, the Zermelo-Fraenkel axioms constitute the standard axiomatization of set theory. They are as follows:

Extensionality:  Two sets are the same if they have the same elements.

 Regularity:  Every non-empty set S contains a member x disjoint from S (as a set).

Separation:  If S is a set and p a property posessable by elements of S, then there is a subset S’ of S containing exactly those elements of S with property p.

Pairing:  If S and T are sets then there is a set containing S and T.

Union:  If S is a set, there exists a set T containing every element of all elements of S.

Infinity:  There is a set S, of which the empty set is an element, such that for all x in S, the union of x and {x} is also in S.

Power Set:  If S is a set, there is a set T containing every subset of S.

In addition to these axioms, the axiom of choice is often assumed, stated here in two equivalent forms:

(i) If A is a collection of non-empty sets then there is a set S containing exactly one element from each element of A. 

(ii) Every set S can be well-ordered.

With the axiom of choice included, Zermelo-Fraenkel Set Theory is commonly abbreviated ZFC, and this practice will be adopted here. By Gödel’s second incompleteness theorem, the consistencey of ZFC cannot be proved within the framework of ZFC. Nevertheless, it is commonly accepted that ZFC is consistent, as no inconsistencies have arisen. Thus, it is desirable to prove statements resulting from the axioms of ZFC.

Cardinal Numbers and the Continuum Hypothesis

Define an equivalence relation ~ between sets as S~T iff there exists a bijection between S and T. An equivalence class under this relation is referred to as the cardinality of its member sets. The cardinality of a set S is abbreviated card(S).  An order relation ≤ is defined for cardinal numbers m and n as m ≤ n if there is an injection of m into n.

Theorem (Cantor):  card(N) < card((0,1))

Proof (Cantor):  Assigin to each number in (0,1) its unique decimal representation, say ai = .ai1ai2ai3... for all i in (0,1).  Place these ai in a countable column beginning with a1. Let b = .a11a22a33....  Then b is in (0,1), but is not in the countable column.  Therefore, for any bijective function from real numbers to natural numbers, there is a real number not in the domain, and card(N) < card((0,1)).Q.E.D.

This is the famous “diagonal method” proof of the uncountability of the reals.  Commonly, א0 := card(N) and c := card((0,1)). It  was proved by Cantor that c = 2א_0.  

The Continuum Hypothesis, proposed by Cantor in 1877, says the following:  

There does not exist a set S such that א0 < card(S) < c.  

The proof or disproof of the Continuum Hypothesis was the first problem posed by David Hilbert at the International Congress of Mathematicians in 1901. Though Cantor at one time believed that he possessed a proof of his hypothesis, this was proved impossible (within ZFC) by Gödel and Cohen. Thus, the search for a new axiom that would confirm or negate the Continuum Hypothesis began. 

The Continuum Hypothesis follows from Gödel’s Axiom of Constructibility, but this axiom is widely believed to be too restrictive and probably false. Conversely, the Continuum Hypothesis implies Martin’s Axiom, and under Zermelo-Fraenkel Set Theory the Generalized Continuum Hypothesis implies the axiom of choice. This of course is very weak evidence, if evidence at all, for the Continuum Hypothesis. The mathematical community is still split over the issue.

Freiling’s Axiom of Symmetry

Chris Freiling proposed in 1986 an axiom that is equivalent to the negation of the Continuum Hypothesis:  his Axiom of Symmetry. This, he believed, would settle the debate. Freiling’s Axiom of Symmetry is as follows:

Let A be the set of all functions from (0,1) to countable subsets of (0,1). Then for all f in A, there exist x and y in (0,1) such that x is not in f(y) and y is not in f(x).

That this statement is equivalent to the negation of the Continuum Hypothesis was first proved by Sierpiński. I have not encountered this proof, and instead offer my own:

Theorem:  Under ZFC, Freiling’s Axiom of Symmetry is true iff the Continuum Hypothesis is false.

Proof:  (=>) Suppose the Continuum Hypothesis is true. Then card((0,1)) = א1. Well-order the elements of (0,1) as r1,r2,r3,… and let f(ri) = {rj: j ≤ i}. Then f is in A, and for j ≤ i, rj is in f(ri), for j > i, ri is in f(rj). Furthermore, f(ri) is countable for all i (having cardinality less than א1 => Freiling’s Axiom of Symmetry is false.

(<=) Suppose the Continuum Hypothesis is false. Then there exists a set S with א0 < card(S) < c. Say card(S) = Ω. Let f be in A. Let B be a set in (0,1) of cardinality Ω. Then f(B) has cardinality א0*Ω = Ω => (0,1)\f(B) has cardinality c. Let C be a set in (0,1)\f(B) of cardinality א0. Then card(f(C)) = א02 = א0 => B\f(C) has cardinality Ω. Let D = B\f(C). Then C and D are nonempty sets such that C∩f(D) and D∩f(C) are empty. This affirms Freiling’s Axiom of Symmetry.Q.E.D. 

Freiling’s Case

The intuition behind Freiling’s Axiom is as follows:  If we fix a random real number x, and assign to x a countable set of real numbers f(x) by some rule f, and then choose a second real number y at random, the probability that y will lie in f(x) is 0. This is because the set f(x) has Lebesgue measure zero. Therefore, we may reasonably predict that y is not in f(x). Since this is true for all x, Freiling infers that we may make our prediction before fixing x. That is, Freiling proposes that with probability 1, if we choose x and then choose y at random, y will not lie in f(x). Furthermore, the order of x and y should not matter, so with probability 1, Freiling proposes, x will also not lie in f(y). Freiling proposes that since this situation should occur with probability 1, we may infer that we can at least find one pair {x,y} for any f with x not in f(y) and y not in f(x). This seems at first to be a reasonable proposition, and is at very least more intuitive than the Continuum Hypothesis. This, of course, is nowhere near a proof that the Continuum Hypothesis is false. 

Nevertheless, Freiling believes that his axiom is derived from intuitive philosophical principles. Namely:

•	We may choose reals at random.
•	A random real will predictably not be contained in a fixed Lebesgue measure zero set.
•	If an accurate prediction can always be made after an event occurs (i.e. the first random real number x is chosen) and no matter what the outcome of the event (for any x) the prediction is the same, then the prediction is accurate if made before the event.
•	The order that random real numbers are chosen should not effect the outcome of Freiling’s experiment.

Again, these seem to be reasonable propositions, and intuition suggests that they are true. 

The Case Against Freiling’s Axiom

Since the probability experiment that Freiling proposes deals with non-Lebesgue measurable sets, Freiling’s “probability 1” outcome is already not well-defined under formal treatment. Nevertheless, his axiom at least seems true; I must admit that after some initial reflection I had convinced myself that Freiling’s Axiom of Symmetry is true. The weakness in Freiling’s intuition only reveals itself in the constructive proof that the Continuum Hypothesis implies the negation of Freiling’s axiom. That under the Continuum Hypothesis we may construct a function that negates the Axiom of Symmetry is a strong indication that the intuition of the axiom may be flawed (as intuition so commonly is). Stated differently, the Continuum Hypothesis provides powerful evidence against Freiling’s intuition, so that one cannot presently decide to support Freiling’s axiom without having already decided to negate the Continuum Hypothesis. If one considers only the Continuum Hypothesis and Freiling’s Axiom of Symmetry in the context of ZFC, then, it is clear that any conclusion about Freiling’s axiom must be based in an initial conclusion about the Continuum Hypothesis, and not the other way around, because the intuition behind Freiling’s axiom is demonstratably weaker than that required for a true axiom. That slight revisions of the Axiom of Symmetry contradict the axiom of choice need not be considered; from the prior arguments it is plain that Freiling’s axiom can be no indication of the truth factor of the Continuum Hypothesis. 

References

Levy, Azriel. Basic Set Theory. Springer-Verlag, London, 1979.

Borsuk, Sierpińsky. Fundamenta Mathematicae. v. 35-36. Warsaw, 1948.

Freiling, Chris. Axioms of Symmetry:  Throwing Darts at the Real Number Line. 1986.

Machover, Moshé. Set Theory, Logic, and their Limitations. University Press, Cambridge, 1996.

Sierpińsky, Wacław. Cardinal and Ordinal Numbers. Polish Scientific Publishers, Warsaw, 1965.