Devlin's Angle

June 2001

How many real numbers are there?

How many real numbers are there? One answer is, "Infinitely many." A more sophisticated answer is "Uncountably many," since Georg Cantor proved that the real line -- the continuum -- cannot be put into one-one correspondence with the natural numbers. But can we be more precise?

Cantor introduced a system of numbers for measuring the size of infinite sets: the alephs. The name comes from the symbol Cantor used to denote his infinite numbers, the Hebrew letter aleph -- a symbol not universally available for web pages. He defined an entire infinite hierarchy of these infinite numbers (or cardinals), aleph-0 (the first infinite cardinal, the size of the set of natural numbers), aleph-1 (the first uncountable cardinal), aleph-2, etc.

The infinite cardinals can be added and multiplied, just as the finite natural numbers can, only it's much easier to learn the answers. The sum or product of any two infinite cardinals is simply the larger of the two.

You can also raise any finite or infinite cardinal to any finite or infinite cardinal power. And this is where things rapidly become tricky. To pick the simplest tricky case, if K is an infinite cardinal, what is the value of 2K (2 raised to the power K)? Cantor proved that the answer is strictly bigger than K itself, but that's as far as he got. In particular, he was not able to figure out whether or not 2(aleph-0) is equal to aleph-1.

The significance of this question for the rest of mathematics lay in the fact that 2(aleph-0) is the size of the real continuum, i.e., the number of real numbers. Since Cantor had been able to prove that there are aleph-0 rational numbers, the next obvious question to ask was, how many real numbers are there? Being unable to answer this question was frustrating to say the least, and Hilbert included the problem in his famous 1900 list.

The proposal that 2(aleph-0) = aleph-1 became known as Cantor's Continuum Hypothesis. It turned out to be intimately connected with the choice of axioms for the construction of infinite sets. The axioms generally accepted by the mathematical community were formulated by Ernst Zermelo and Abraham Fraenkel in the early twentieth century. In 1936, Kurt Goedel stunned the mathematical world with his proof that the Zermelo-Franekel axioms were not sufficient to prove that the Continuum Hypothesis is false.

What made this stunning was not the result itself. Apart from logicians and a few real analysts, most mathematicians didn't care about the Continuum Hypothesis one way or the other. Rather, it was the fact that Goedel had found a way to prove, conclusively, that something could not be proved. (Notice that Goedel's proof that the Continuum Hypothesis could not be proved in Zermelo-Fraenkel set theory did not imply that it could be disproved in that theory. Absence of proof -- even proven absence of proof -- is not proof of the contrary.)

With the sure knowledge that the Continuum Hypothesis could not be proved false, the hunt was on to prove it true. That hunt proved unfruitful, and in 1963 Paul Cohen showed why. In a mathematical tour de force that won him a Fields Medal, he proved that the Continuum Hypothesis could not be proved true either! (Within the axiomatic framework of Zermelo and Fraenkel.) The hypothesis was undecidable.

Of course, the natural response was to look for additional axioms of set theory to augment the Zermelo-Fraenkel system, that would enable the Continuum Hypothesis to be resolved one way or the other. And many mathematicians did just that. But without success.

The problem was that set theory was a foundational subject, one that had been developed in an attempt to provide a unified framework for all of mathematics (including arithmetic). Its axioms, to be acceptable, had to be "intuitively obvious." No one could find such a principle.

One possibility that I personally found appealing (my Ph.D was in set theory and Cantor's infinite cardinal arithmetic, and I specialized in that area for the first fifteen years of my career) was the "Axiom of Constructibility". This principle was formulated by Goedel in the course of his proof that the Continuum Hypothesis could not be disproved using the Zermelo- Fraenkel axioms. Although Goedel did not propose adopting it as an axiom of set theory, I felt it had sufficient "naturalness" in its favor to do so. Not because I believed it was "true." When it comes to doing mathematics on infinite sets, I don't think the notion of truth comes into it. Rather, I felt that the meta-message in Cohen's result (and a lot of similar results that came in its wake) was that the axioms of set theory should be chosen on pragmatic grounds.

On the basis of set theory having the main goal of providing a universal basis for mathematics, I could (and in 1977 did) put forward what I believed was a good argument in favor of adopting the Axiom of Constructibility. (I laid out my argument in my monograph The Axiom of Constructibity: A Guide for the Mathematician, published in the Springer-Verlag Lecture Notes in Mathematics series in 1977.)

If the Axiom of Constructibility was assumed (as an additional axiom, on top of the Zermelo-Fraenkel system), then you could prove that the Continuum Hypothesis is true.

For a variety of reasons, many mathematicians did not buy my arguments, or those of others who also proposed the Axiom of Constructibility. But no one came up with what I thought was a compelling counter argument. At least, not at the time. That changed in 1986, when Christopher Freiling published an intriguing paper in Volume 51 of the Journal of Symbolic Logic. In his paper, titled "Axioms of Symmetry: throwing darts at the real line", Freiling puts forward the following thought experiment.

You and I are throwing darts at a dartboard. We are separated by a screen, so that nothing either of us does can influence the other. At a given signal from a third party, we both throw a dart at the board. We do so entirely randomly. (Formally, since the points on the dartboard can be put into a one-one correspondence with the real numbers, we are simply two independent random number generators.)

How is the winner decided? Well, the organizer has chosen a well-ordering of the real numbers (i.e., the points on the dartboard), say <<. The aim is to land on a point associated with the larger number. If your dart lands on a number (point) Y that is << the number M my dart lands on, I win; otherwise you win. Simple, no?

Well, there's more. Suppose the Continuum Hypothesis were true. Then the organizer could have chosen the well-ordering so that, for any number X, the set {R|R << X} is countable. Agreed?

Now, since we throw independently, we can assume I threw first. My dart lands at point M. Now you throw. Since the set {R|R << M} is countable, the probability that your dart lands at a point Y for which Y << M is zero. (Any countable set has measure zero.) Thus, with probability 1, Y>> M, and you win.

But the situation is entirely symmetrical, and so by the same argument, with probability 1, I win.

But this is an impossible situation. Conclusion: there can be no such well ordering, and hence the Continuum Hypothesis is false. Right?

Well, not quite. To make the above argument go through formally, we have assume that the graph of the well ordering << is measurable. And there is no justification for making such an assumption. Thus, we have not proved that the Continuum Hypothesis is false. But that was not what we (or Freiling) were trying to do. Rather, we were looking for some plausible evidence to support an axiomatization of set theory that would resolve the Continuum Hypothesis.

If you view set theory as an axiomatic framework for constructing sets, and take a conservative approach of constructing only the sets that the rest of mathematics absolutely must have, you end up with the Axiom of Constructibility, and then the Continuum Hypothesis is true. But if you conceive of mathematics as abstracting from the world of our everyday experience, and if you take the view that Freiling's dartboard thought experiment has an intuitive naturalness and "ought to be right", then your set theory, whatever its axioms may be, should imply that the Continuum Hypothesis is false. (Or at the very least, your axioms should not imply that the Continuum Hypothesis is true.)

What's my own current view? Well, I still think a good argument can be put forward for the Axiom of Constructibility. But I also find the Freiling thought experiment compelling. To my mind, on an intuitive level, it does show that the Continuum Hypothesis must be false. When a mathematician finds himself supporting two contradictory propositions, he's obviously been a department chair or a dean for too long and it's time to give up and move on. And do you know, I just did. Please note the change of address below.


Devlin's Angle is updated at the beginning of each month.
Keith Devlin devlin@csli.stanford.edu) is the new Executive Director of the Center for the Study of Language and Information at Stanford University, and "The Math Guy" on NPR's Weekend Edition. His latest book is The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, published by Basic Books.