In 1900, David Hilbert published a list of twenty-three open questions in mathematics, ten of which he presented at the International Congress of Mathematics in Paris that year. Hilbert had a good nose for asking mathematical questions as the ones on his list went on to lead very interesting mathematical lives. Many have been solved, but some have not been, and seem to be quite difficult. In both cases, some very deep mathematics has been developed along the way. The so-called Riemann hypothesis, for example, has withstood the attack of generations of mathematicians ever since 1900 (or earlier). But the effort to solve it has led to some beautiful mathematics. Hilbert’s fifth problem turned out to assert something that couldn’t be true, though with fine tuning the “right” question—that is, the question Hilbert should have asked—was both formulated and solved. There is certainly an art to asking a good question in mathematics.
The problem known as the continuum hypothesis has had perhaps the strangest fate of all. The very first problem on the list, it is simple to state: how many points on a line are there? Strangely enough, this simple question turns out to be deeply intertwined with most of the interesting open problems in set theory, a field of mathematics with a very general focus, so general that all other mathematics can be seen as part of it, a kind of foundation on which the house of mathematics rests. Most objects in mathematics are infinite, and set theory is indeed just a theory of the infinite.
How ironic then that the continuum hypothesis is unsolvable—indeed, “provably unsolvable,” as we say. This means that none of the known mathematical methods—those that mathematicians actually use and find legitimate—will suffice to settle the continuum hypothesis one way or another. It seems odd that being unsolvable is the kind of thing one can prove about a mathematical question. In fact, there are many questions of this type, particularly about sets of real numbers—or sets of points on a line, if you like—that we know cannot be settled using standard mathematical methods.
Now, mathematics is not frozen in time or method—to the contrary, it is a very dynamic enterprise, each generation expanding and building on what went before. This process of expansion has not always been easy; sometimes it takes a while before new methods are accepted. This was true of set theory in the late nineteenth century. Its inventor, Georg Cantor, met with serious opposition on the part of those who were hesitant to admit infinite objects into mathematics.
What concerns us here is not so much the prehistory of the continuum hypothesis, but the present state of it, and the remarkable fact that mathematicians are in the midst of developing new methods by which the continuum hypothesis could be solved after all.
I will explain some of these developments, along with some of the more recent history of the continuum hypothesis, from the point of view of Kurt Gödel’s role in them. Gödel, a Member of the Institute’s School of Mathematics on several occasions in the 1930s, and then continuously from 1940 until 1976,1 was a relative newcomer to the problem. But it turns out that Gödel’s hand is visible in virtually every aspect of the problem, from the post-Cantorian period onward. Curiously enough, this is even more true now than it was at the time of Gödel’s death nearly thirty-five years ago.
What is the Continuum Hypothesis?
Mathematics is nowadays saturated with infinity. There are infinitely many positive whole numbers 0, 1, 2, 3 . . . . There are infinitely many lines, squares, circles in the plane, balls, cubes, polyhedra in the space, and so on. But there are also different degrees of infinity. Let us say that a set—a collection of mathematical objects such as numbers or lines—is countable if it has the same number of elements as the sequence of positive whole numbers 1, 2, 3 . . . . The set of positive whole numbers is thus countable, and so is the set of all rational numbers. In the early 1870s, Cantor made a momentous discovery: the set of real numbers (such as 5, 17, 5/12, √–2, π, e, . . . ) sometimes called the “continuum,” is uncountable. By uncountable, we mean that if we try to count the points on a line one by one, we will never succeed, even if we use all of the whole numbers. Now it is natural to ask the following question: are there any infinities between the two infinities of whole numbers and of real numbers?
This is the continuum hypothesis, which proposes that if you are given a line with an infinite set of points marked out on it, then just two things can happen: either the set is countable, or it has as many elements as the whole line. There is no third infinity between the two.
At first, Cantor thought he had a proof of the continuum hypothesis; then he thought he could prove it was false; and then he gave up. This was a blow to Cantor, who saw this as a defect in his work—if one cannot answer such a simple question as the continuum hypothesis, how can one possibly go forward?
The continuum hypothesis went on to become a very important problem, so much so that in 1900 Hilbert listed it as the first on his list of open problems, as previously mentioned. Hilbert eventually gave a proof of it in 1925—the proof was wrong, though it contained some important ideas.
Around the turn of the century, mathematicians were able to prove that the continuum hypothesis holds for a special class of sets called the Borel sets.2 This is a concrete class of sets, containing, for the most part, the usual sets that mathematicians work with. Even with this early success in the special case of Borel sets though, and in spite of Hilbert’s attempted solution, mathematicians began to speculate that the continuum hypothesis was in general not solvable at all. Hilbert, for whom nothing less than “the glory of human existence” seemed to depend upon the ability to resolve all such questions, was an exception. “Wir müssen wissen. Wir werden wissen,”3 he said in 1930 in Königsberg. In a great irony of history, at the very same meeting, but on the day before, the young Gödel announced his first incompleteness theorem. This theorem, together with Gödel’s second incompleteness theorem, is generally thought to have dealt a death blow to Hilbert’s idea that every mathematical question that permits an exact formulation can be solved. Hilbert was not in the room at the time.
Gödel, however, became a strong advocate of the solvability of the continuum hypothesis, taking the view that his incompleteness theorems, though they show that some provably undecidable statements do exist, have nothing to do with whether the continuum hypothesis is solvable or not. Like Hilbert, Gödel maintained that the continuum hypothesis will be solved.
What is Provable Unsolvability Anyway?
We arrive at an apparent conundrum. On the one hand, the continuum hypothesis is provably unsolvable, and on the other hand, both Gödel and Hilbert thought it was solvable. How to resolve this difficulty? What does it mean for something to be provably unsolvable anyway?
Some mathematical problems may be extremely difficult and therefore without a solution up to now, but one day someone may come up with a brilliant solution. Fermat’s last theorem, for example, went unsolved for three and a half centuries. But then Andrew Wiles was able to solve it in 1994. The continuum hypothesis is a problem of a very different kind; we actually can prove that it is impossible to solve it using current methods, which is not a completely unknown phenomenon in mathematics. For example, the age-old trisection problem asks: can we trisect a given angle by using just a ruler and compass? The Greeks of the classical period were very puzzled by how to make such a trisection, and no wonder, for in the nineteenth century it was proved that it is impossible—not just very difficult but impossible. You need a little more than a ruler and compass to trisect an arbitrary angle—for example, a compass and a ruler with two marks on it.
It is the same with the continuum hypothesis: we know that it is impossible to solve using the tools we have in set theory at the moment. And up until recently nobody knew what the analogue of a ruler with two marks on it would be in this case. Since the current tools of set theory are so incredibly powerful that they cover all of existing mathematics, it is almost a philosophical question: what would it be like to go beyond set-theoretic methods and suggest something new? Still, this is exactly what is needed to solve the continuum hypothesis.
Gödel began to think about the continuum problem in the summer of 1930, though it wasn’t until 1937 that he proved the continuum hypothesis is at least consistent. This means that with current mathematical methods, we cannot prove that the continuum hypothesis is false.
Describing Gödel’s solution would draw us into unneeded technicalities, but we can say a little bit about it. Gödel built a model of mathematics in which the continuum hypothesis is true. What is a model? This is something mathematicians build with the purpose of showing that something is possible, even if we admit that the model is just what it is, a kind of artificial construction. Children build model airplanes; architects draw up architectural plans; mathematicians build models of the mathematical universe. There is an important difference though, between mathematicians’ models and architectural plans or model airplanes: building a model that has the exact property the mathematician has in mind, is, in all but trivial cases, extremely difficult. It is like a very great feat of engineering.
The idea behind Gödel’s model, which we now call the universe of constructible sets, was that it should be made as small as is conceivably possible by throwing everything out that was not absolutely essential. It was a tour de force to show that what was left was enough to satisfy the requirements of mathematics, and, in addition, the continuum hypothesis. This did not show that the continuum hypothesis is really true, only that it is consistent, because Gödel’s universe of constructible sets is not the real universe, only a kind of artifact. Still, it suffices to demonstrate the consistency of the continuum hypothesis.
After Gödel’s achievement, mathematicians sought a model in which the continuum hypothesis fails, just as Gödel found a model in which the continuum hypothesis holds. This would mean that the continuum hypothesis is unsolvable using current methods. If, on the one hand, one can build a picture of the mathematical universe in which it is true, and, on the other hand, if one can also build another universe in which it is false, it would essentially tell you that no information about the continuum hypothesis is lurking in the standard machinery of mathematics.
So how to build a model for the failure of the continuum hypothesis? Since Gödel’s universe was the only nontrivial universe that had been introduced, and, moreover, it was the smallest possible, mathematicians quickly realized that they had to find a way to extend Gödel’s model, by carefully adding real numbers to it. This is hair-raisingly difficult. It is like adding a new card to a huge house of cards, or, more exactly, like adding a new point to a line that already is—in a sense—a continuum. Where do you find the space to slip in a few new real numbers?
Looking back at Paul Cohen’s solution, a logician has to slap her forehead, not once, but a few times. His idea was that the real numbers one adds should have “no properties,” as strange as this may sound; they should be “generic,” as he called them. In particular, a Cohen real, as they came to be called, should avoid “saying anything” nontrivial about the model. How to make this idea mathematically precise? That was Paul Cohen’s great invention: the forcing method, which is a way to add new reals to a model of the mathematical universe.
Even with this idea, serious obstacles now stood in the way of a full proof. For example, one has to prove an extremely delicate metamathematical theorem—as these are called—that even though forcing extends the universe to a bigger one, one can still talk about it in the first universe; in technical terms, one has to prove that forcing is definable. Moreover, to violate the continuum hypothesis, we have to add a lot of new points to the continuum, and what we believe is “a lot” may in the final stretch turn out to be not so many after all. This last problem—the technical term is preserving cardinals—was a very serious matter. Cohen later wrote of his sense of unease at that point, “given the rumors that had circulated that Gödel was unable to handle the CH.”4 Perhaps Cohen sensed, while on the brink of his great discovery, the almost physical presence of the one mathematician who had walked the very long way up to that very door, but was unable to open it.
Two weeks later, while vacationing with his family in the Midwest, Cohen suddenly remembered a lemma from topology (due to N. A. Shanin), and this was just what was needed to show that everything falls into place. The proof was now finished. It would have been an astounding achievement for any set theorist, but the fact that it was solved by someone from a completely different field—Paul Cohen was an analyst after all, not a set theorist—seemed beyond belief.
Writing the Paper
The story of what happened in the immediate aftermath of Cohen’s announcement of his proof is very interesting, also from the point of view of human interest, so we will permit ourselves a slight digression in order to touch upon it here.
The announcement seems to have been made at a time when the extent of what had been shown was not clear, and the proof, though it was finished in all the essentials, was not in all details completely finished. In a first letter to Gödel, dated April 24, 1963, Cohen communicated his results. But about a week later, he wrote a second, more urgent letter, in which he expressed his fear that there might be a hidden flaw in the proof, and, at the same time, his exasperation with logicians, who could not believe that he was able to prove that very delicate theorem on the definability of forcing.
Cohen confessed in the letter that the situation was wearing, also considering “the unexpected interest my work has aroused among the general (non-logical) mathematical world.”
Gödel replied with a very friendly letter, inviting Cohen to visit him, either at his home on Linden Lane or in his office at the Institute, writing, “You have just achieved the most important progress in set theory since its axiomatization. So you have every reason to be in high spirits.”
Soon after receiving the letter, Cohen visited Gödel at home, whereupon Gödel checked the proof, and pronounced it correct.
What followed over the next six months is a voluminous correspondence between the two, centered around the writing of the paper for the Proceedings of the National Academy of Sciences. The paper had to be carefully written; but Cohen was clearly impatient to go on to other work. It therefore fell to Gödel to fine tune the argument, as well as simplify it, all the while keeping Cohen in good spirits. The Gödel that emerges in these letters—sovereign, generous, and full of avuncular goodwill, will be unfamiliar to readers of the biographies—especially if one keeps in mind that by 1963 Gödel had devoted a good part of twenty-five years to solving the continuum problem himself, without success. “Your proof is the very best possible,” Gödel wrote at one point. “Reading it is like reading a really good play.”
Gödel and Cohen bequeathed to set theorists the only two model construction methods they have. Gödel’s method shows how to “shrink” the set-theoretic universe to obtain a concrete and comprehensible structure. Cohen’s method allows us to expand the set-theoretic universe in accordance with the intuition that the set of real numbers is very large. Building on this solid foundation, future generations of set theorists have been able to make spectacular advances.
There was one last episode concerning Gödel and the continuum hypothesis. In 1972, Gödel circulated a paper called “Some considerations leading to the probable conclusion that the true power of the continuum is ℵ2,” which derived the failure of the continuum hypothesis from some new assumptions, the so-called scale axioms of Hausdorff. The proof was incorrect, and Gödel withdrew it, blaming his illness. In 2000, Jörg Brendle, Paul Larson, and Stevo Todorcevic5 isolated three principles implicit in Gödel’s paper, which, taken together, put a bound on the size of the continuum. And subsequently Gödel’s ℵ2 became a candidate of choice for many set theorists, as various important new principles from conceptually quite different areas were shown to imply that the size of the continuum is ℵ2.
Currently, there are two main programs in set theory. The inner model program seeks to construct models that resemble Gödel’s universe of constructible sets, but such that certain strong principles, called large cardinal axioms, would hold in them. These are very powerful new principles, which go beyond current mathematical methods (axioms). As Gödel predicted with great prescience in the 1940s, such cardinals have now become indispensable in contemporary set theory. One way to certify their existence is to build a model of the universe for them—not just any model, but one that resembles Gödel’s constructible universe, which has by now become what is called “canonical.” In fact, this may be the single most important question in set theory at the moment—whether the universe is “like” Gödel’s universe, or whether it is very far from it. If this question is answered, in particular if the inner model program succeeds, the continuum hypothesis will be solved.
The other program has to do with fixing larger and larger parts of the mathematical universe, beyond the world of the previously mentioned Borel sets. Here also, if the program succeeds, the continuum hypothesis will be solved.
We end with the work of another seminal figure, Saharon Shelah. Shelah has solved a generalized form of the continuum hypothesis, in the following sense: perhaps Hilbert was asking the wrong question! The right question, according to Shelah, is perhaps not how many points are on a line, but rather how many “small” subsets of a given set you need to cover every small subset by only a few of them. In a series of spectacular results using this idea in his so-called pcf-theory, Shelah was able to reverse a trend of fifty years of independence results in cardinal arithmetic, by obtaining provable bounds on the exponential function. The most dramatic of these is 2ℵω ≤ 2ℵ0 + ℵω4 . Strictly speaking, this does not bear on the continuum hypothesis directly, since Shelah changed the question and also because the result is about bigger sets. But it is a remarkable result in the general direction of the continuum hypothesis.
In his paper,6Shelah quotes Andrew Gleason, who made a major contribution to the solution of Hilbert’s fifth problem:
Of course, many mathematicians are not aware that the problem as stated by Hilbert is not the problem that has been ultimately called the Fifth Problem. It was shown very, very early that what he was asking people to consider was actually false. He asked to show that the action of a locally-euclidean group on a manifold was always analytic, and that’s false . . . you had to change things considerably before you could make the statement he was concerned with true. That’s sort of interesting, I think. It’s also part of the way a mathematical theory develops. People have ideas about what ought to be so and they propose this as a good question to work on, and then it turns out that part of it isn’t so.
So maybe the continuum problem has been solved after all, and we just haven’t realized it yet.
1Appointed to the permanent Faculty in 1953; 2 This was extended to the so-called analytic sets by Mikhail Suslin in 1917. Borel sets are named for Emile Borel, uncle of the late mathematician (and IAS Faculty member) Armand Borel.; 3 “We must know. We will know.”; 4 P. J. Cohen, “The Discovery of Forcing”; 5 In their "Rectangular Axioms, Perfect Set Properties and Decomposition"; 6 "The Generalized Continuum Hypothesis Revisited"
Some Mathematical Details
Intuitively, the set-theoretic universe is the result of iterating basic constructions such as products ∏i∈IAi, unions Ui∈I Ai, and power sets P(A). In addition, the universe is assumed to satisfy so-called reflection: any property that it has is already possessed by some smaller universe, the domain of which is a set. The process starts from some given urelements, objects that are not sets, i.e., do not consist of elements, but it has been proven that the urelements are unnecessary and the process can be started from the empty set. Iterating this process into the transfinite, we obtain the cumulative hierarchy V of sets. Transfinite iterations are governed by ordinals, canonical representatives of well-ordered total orders, denoted by lower-case Greek letters α, β, etc. The hierarchy V is defined recursively by Vα = Uβ <α P(Vβ). The fact that V = Uα Vα is the entire universe of sets is the intuitive content of the axioms of Zermelo-Frankel set theory with the Axiom of Choice, or ZFC, the basic system we have been working with all along.
Now Gödel’s model of the ZFC axioms, the constructible hierarchy L = Uα Lα, where Lα = Uβ<α PL(Vβ), is built up not by means of the unrestricted power set operation P(A), but by the restricted operation PL(A), which takes from P(A) only those sets that are definable in (A, ∈). Gödel showed that we can consistently assume V = L, but Cohen showed that it is consistent to assume that there are real numbers that are not in L.
The Borel sets of reals are obtained from open sets by means of iterating complements and countable unions. If we enlarge the set of Borel sets by including images of continuous functions, we obtain the analytic sets; a set is coanalytic if its complement is analytic.
Finally, the projective sets are obtained from analytic sets by iterating complements and continuous images. The field of descriptive set theory asks, among other questions, whether the classical theory of analytic and coanalytic sets can be extended to the projective sets; in particular, whether the projective sets are Lebesgue measurable, and have the perfect set property and the property of Baire. This was settled in the 1980s with the work of Shelah and Woodin, building on earlier work of Solovay, who showed that the projective sets have these three properties as a consequence of the existence of certain so-called large cardinals. This also follows from projective determinacy, a principle that was shown by Martin and Steel to follow from the existence of such large cardinals. A cardinal α is called a large cardinal if Vα behaves in certain ways like V itself. For example, in that case, Vα is a model of ZFC, but more is assumed. A famous large cardinal is a measurable cardinal, introduced by Stanislaw Ulam, an example of which is the smallest cardinal that admits a nontrivial countably additive two-valued measure.
What a State Mathematics Would Be In Today . . .
Before coming to the Institute where he was appointed as one of its first Professors in 1933, John von Neumann was a student of David Hilbert’s in Göttingen. Von Neumann worked on Hilbert’s program to find a complete and consistent set of axioms for all of mathematics. In addition to his many other contributions to mathematics and physics, von Neumann defined Hilbert space (unbounded operators on an infinite dimensional space), which he used to formulate a mathematical structure of quantum mechanics. Below, the late Herman Goldstine, a former Member in the Schools of Mathematics, Natural Sciences, and Historical Studies, recalls von Neumann’s working dreams about Kurt Gödel’s incompleteness theorem(s). (Excerpted from an oral history transcript available at www.princeton.edu/%7Emudd/finding_aids/mathoral/pmc15.htm; more information about von Neumann and Gödel is available at www.ias.edu/people/noted-figures.)
His work habits were very methodical. He would get up in the morning, and go to the Nassau Club to have breakfast. And then from the Nassau Club he’d come to the Institute around nine, nine-thirty, work until lunch, have lunch, and then work until, say, five, and then go on home. Many evenings he would entertain. Usually a few of us, maybe my wife and me. We would just sit around, and he might not even sit in the same room. He had a little study that opened off of the living room, and he would just sit in there sometimes. He would listen, and if something interested him, he would interrupt. Otherwise he would work away.
At night he would go to bed at a reasonable hour, and he would waken, I think, almost every night, judging from the things he told me and the few times that he and I shared hotel rooms. He would waken in the night, two, three in the morning, and would have thought through what he had been working on. He would then write. He would write down the things he had worked on. . . .
He, under Hilbert’s tutelage, was trying to prove the opposite of the Gödel theorem. He worked and worked and worked at this, and one night he dreamed the proof. He got up and wrote it down, and he got very close to the end. He went and worked all day on that part, and the next night he dreamed again. He dreamed how to close the gap, and he got up and wrote, and he got within epsilon of the end, but he couldn’t make the final step. So he went to bed. The next day he worked and worked and worked at it, and he said to me, “You know, it was very lucky, Herman, that I didn’t dream the third night, or think what a state mathematics would be in today.” [Laughter.]