I'd like to tell you about a certain confounding mystery at the center of set theory, concerned with Cantor's continuum hypothesis. We saw how Cantor proved that the set of real numbers is an uncountable infinity, and therefore strictly larger than the set of natural numbers. We have, on one hand, the countable infinity of the natural numbers, and on the other, the uncountable infinity of the real numbers. When you prove that one infinity is bigger than another, it must be the most natural question in the world to ask: is there any infinity in between?
That question is precisely what the continuum hypothesis is concerned with. Cantor knew that the set of real numbers was a bigger infinity than the natural numbers, but he did not know whether there was anything in between them. The continuum hypothesis, as he formulated it, is the assertion that there is nothing in between. It was, however, an open question for Cantor. He became obsessed with it, spending his entire life trying to prove it, trying to refute it, and failing at both. He never knew the answer, and died before one was found. The answer, when it finally came, is quite a remarkable story that unfolded over the decades that followed.
Cantor had laid out a strategic program for attempting to prove that the continuum hypothesis is true, that is, that no infinity lies between the natural numbers and the real numbers. His approach was to establish the result first for very simple kinds of sets, and then work his way up to more and more complex ones. He succeeded for closed sets of reals, and the result holds for open sets as well. The plan was to advance through successive levels of complexity, showing at each stage that no set of that kind could have a cardinality intermediate between the naturals and the reals. Regrettably, he died before reaching a conclusion, though his program was continued to a certain extent by others, as I will discuss shortly.
The eminent mathematician David Hilbert placed the continuum hypothesis at the very top of his famous list of open problems, presented at the dawn of the twentieth century, which went on to guide so much of mathematical research in the decades that followed. The continuum problem was number one on that list. Nobody knew whether it was true or false, and no one could produce a set of intermediate cardinality. There were not even any candidate sets to examine.
Consider what such a set would have to look like. You would need a set of real numbers that is not countable, since it must be larger than the natural numbers, but that is also not equinumerous with the full set of real numbers. It would have to be genuinely intermediate in size, sitting strictly between those two infinite cardinalities. The question is whether any such set exists, or whether no such thing is possible. That is exactly what is at stake in the continuum hypothesis.
One can set the stage by observing that equinumerosity is ubiquitous: there are many, many sets that we can prove are equinumerous with one another. In the context of the discussion with Galileo, we mentioned that Galileo himself proved that line segments of different lengths are equinumerous, by means of the one-to-one correspondence provided by a fanning construction. He also showed that a given line segment is equinumerous with the whole real line. A different way of seeing this is to consider the arctangent function, which maps the entire real line bijectively onto a bounded open interval, say, from −π/2 to π/2, thereby demonstrating that those two sets have the same size.
There are many further instances of such equinumerosity. For example, the power set of the natural numbers is equinumerous with Cantor space, the set of all binary sequences, and that in turn is equinumerous with the real line. All of these sets are therefore equinumerous with one another. Let us now turn to another very surprising result due to Cantor.
Cantor considered the question whether the real line might be equinumerous with the real plane, that is, with ℝ², the set of pairs of real numbers (x, y). On the one hand, the plane seems to have far more points than the line, since it is two-dimensional. Lines sit inside the plane, but they take up the tiniest fraction of it, having zero area. It would seem reasonable to suppose that there are strictly more points in the plane than on the line.
Amazingly, Cantor observed that the real line and the real plane are in fact equinumerous: they have exactly the same size. When he discovered this, he wrote immediately to Richard Dedekind, this was in 1877, and said, "I see it, but I don't believe it." It is a genuinely incredible and counterintuitive result. There are exactly the same number of points on the line as in the plane, and the same is true in any higher dimension: three, four, five, any finite dimension, and even countably infinite dimension if one considers infinite-dimensional spaces.
To prove that the line and the plane are equinumerous, it suffices to establish two facts. First, there are at least as many points in the plane as on the line, since the line sits inside the plane: the x-axis is a subset of ℝ². Second, one must find a one-to-one mapping from the plane into the line. Together with the Cantor–Schroeder–Bernstein theorem, these two facts imply that the line and the plane have the same cardinality. The second fact is exactly what Cantor established, and what we now demonstrate.
Given any two real numbers, say the x-coordinate is 3.14159265… and the y-coordinate is 2.71828182…, we associate this pair with a single real number by interleaving their digits. We take the first digit of x, then the first digit of y, then the second digit of x, then the second digit of y, and so on, assembling them into one real number. This defines a mapping from pairs of real numbers into the real numbers, and the claim is that it is one-to-one.
One technical point must be addressed: some real numbers have two decimal representations, one ending in an infinite string of nines and another ending in an infinite string of zeros. By convention, we always use the eventually-all-zeros representation for any number where this ambiguity arises. With this convention, the interleaved number we produce will never itself end in all nines, since that would have required both input numbers to end in all nines, a case we have excluded.
Given the combined interleaved number, one can recover x and y uniquely by unraveling: take the digits in the odd positions to recover x, and the digits in the even positions to recover y. This shows the mapping is one-to-one. It is not, however, onto: for instance, a real number with a nine in every other decimal place could never arise as the output of this procedure, since that would have required one of the inputs to be eventually all nines. So the interleaving gives a one-to-one correspondence between ℝ² and a proper subset of ℝ, but that is sufficient to show that the reals are at least as numerous as pairs of reals.
Combining both directions and applying the Cantor–Schroeder–Bernstein theorem, we conclude that the real line is equinumerous with the real plane. The argument extends immediately to higher dimensions. One can interleave the digits of three real numbers just as easily as two, establishing directly that triples correspond one-to-one with reals. Alternatively, once we know that ℝ and ℝ² are equinumerous, we can multiply both sides by ℝ to obtain that ℝ² and ℝ³ are equinumerous as well. Consequently, the line, the plane, and three-dimensional space all have exactly the same number of points, and the same holds in any finite dimension.
Let us ask another question about equinumerosity. If you wanted a large set, would you rather have the set of functions from the reals to the integers, or the set of functions from the integers to the reals? Which is the bigger set? We are comparing the set of functions from the integers to the reals versus the set of functions from the reals to the integers. In exponential notation, XY denotes the set of functions from Y to X, from the exponent to the base. You may want to pause and think about it before reading on.
Consider first the set of functions from the integers to the reals. This set is equinumerous with the set of functions from the natural numbers to the reals, since the integers and the natural numbers have the same cardinality. A function from the natural numbers to the reals is simply an infinite sequence of real numbers, so the question becomes: how many countable sequences of real numbers are there? Using an interleaving argument analogous to the one showing that pairs of natural numbers are equinumerous with the natural numbers, we can interleave the decimal digits of infinitely many real numbers along a winding path to produce a single real number. This argument shows that the set of infinite sequences of real numbers is equinumerous with the reals themselves, so ℝℤ has the same cardinality as ℝ.
Now consider the set of functions from the reals to the integers. This set is at least as large as the set of functions from the reals into the two-element set {0, 1}, that is, the set of {0,1}-valued functions on the reals. But that collection is equinumerous with the power set of the reals, since each such function corresponds to the characteristic function of a subset of ℝ. Therefore ℤℝ is at least as large as the power set of the reals, which by Cantor's theorem is strictly larger than ℝ itself.
The conclusion is that there are vastly more functions from the reals to the integers than there are functions from the integers to the reals. This kind of analysis, surveying the cardinalities of naturally arising sets, is a natural preliminary step when thinking about the continuum hypothesis. One looks at all the sets one knows and asks whether any of them might have a cardinality strictly between that of the natural numbers and that of the reals.
Let us calculate how many elements another natural set contains, namely, how many continuous functions there are on the real numbers. We already know how many functions there are from the reals to the reals in total: there are a great many, strictly more than there are real numbers, because the functions from the reals to the reals include all functions from the reals to a two-element set, and that collection is already equinumerous with the power set of the reals. But many of those functions are not continuous. In fact, the only continuous functions taking just two values must be constant.
Suppose we have a continuous function F from the real numbers to the real numbers. Consider its graph sitting inside the real plane, and look at all the points lying below that graph. The set of points below the graph determines the function entirely, since F is the envelope of that region. Crucially, we do not need all such points; it suffices to record the rational points below the graph. That is, we look at all pairs of rationals (p, q) such that the point lies below the graph of F.
This information is enough to recover F completely. From the rational pairs below the graph, we can determine the value of F at every rational input, since we know precisely which numbers lie below the function at each rational coordinate. Knowing the values at all rational points then determines the values everywhere, because every real number is a limit of rational numbers, and continuity does the rest.
More formally, associate to each continuous function F the set AF consisting of all rational pairs lying below its graph; this is a subset of ℚ × ℚ. The number of continuous functions is therefore at most the number of subsets of ℚ × ℚ. Since ℚ is countable, ℚ × ℚ is equinumerous with ℕ, so the number of subsets of ℚ × ℚ is equinumerous with the power set of ℕ, which is equinumerous with ℝ. Thus the number of continuous functions is bounded above by the cardinality of the reals.
On the other hand, there are at least as many continuous functions as there are real numbers, since for every real number c we have the constant function with value c. Combining the two inequalities, the number of continuous functions from the reals to the reals is exactly equal to the number of real numbers. In particular, most functions, in this equinumerosity sense, are not continuous.
One can become quite adept at computing these equinumerosity classes: judging which sets are equinumerous with one another, which are strictly smaller, and so on. Gradually one builds up a stock of examples, many of which are countable and others equinumerous with the reals. What is striking is that no one has ever defined a specific set of reals that is strictly intermediate in cardinality between the natural numbers and the real numbers, and this is evidence for the continuum hypothesis.
Cantor's strategy for solving the continuum hypothesis was to begin with the simplest sets. One natural class of simple sets is the closed sets, and Cantor proved that every closed set of real numbers is either countable or equinumerous with the entire real line. In other words, no closed set can ever serve as a counterexample to the continuum hypothesis. Let us explain how that argument goes.
A closed set might contain intervals, or it might be highly complicated, or it might contain isolated points. If there are many isolated points in a bounded region, they must have a limit point. Given a closed set C₀, we undertake a procedure called the Cantor–Bendixson process, forming the derivative C₁ by removing all isolated points. Consider a convergent sequence: every point in the sequence sits inside a small open neighborhood containing no other point of the set, so each such point is isolated and gets removed. The limit point, however, is not isolated. Every neighborhood around it contains infinitely many points of the sequence, and so it remains. After one step, only the limit point survives, and in that reduced set it is itself isolated, so it is removed at the next step.
More elaborate examples are possible. Imagine three points, and for each of those points a sequence of white points converging to it, and for each white point a sequence of blue points converging to it. The blue points are all isolated, since they do not interfere with one another, so at the first step of the Cantor-Bendixson process the blue points are removed while the white points remain, as each white point is a limit of blue points and is therefore not isolated. In the resulting set the white points have become isolated, so they are removed at the second step, leaving only the three original points. Those three points are then isolated in turn and are removed at the third step.
One can make such constructions arbitrarily elaborate. Adding red sequences converging to the blue points would extend the process by one further step, and for every finite n one can produce a closed set requiring exactly n steps. By placing side by side a set lasting one step, a set lasting two steps, a set lasting three steps, and so on, one obtains a closed set that requires ω many steps. Adding a limit point to such a configuration produces a set lasting ω + 1 steps, and so on. It was precisely in making sense of this iterated process that Cantor invented the ordinals.
The key observation is that at each stage of the Cantor–Bendixson process, only countably many points can be removed. An isolated point has an open ball around it containing no other point of the set; that ball can be chosen to have a rational center and a rational radius, and since the point was the only member of the set in that ball, no other point will ever claim the same rational ball. Every removed point is thus accounted for by a distinct element of the countable collection of rational balls, so at each stage only countably many points are discarded. From this, Cantor deduced that the process must terminate at some countable transfinite ordinal stage.
It follows that every closed set of real numbers can be written as the union of a countable set, consisting of all the isolated points removed throughout the process, together with a perfect set, that is, a closed set with no isolated points, which is what remains when the process terminates. Cantor had also established that every nonempty perfect set is equinumerous with the entire real line. Therefore, every closed set is either countable, if the Cantor–Bendixson process reaches the empty set and the original set consisted entirely of the countably many discarded isolated points, or equinumerous with the real line, if the process terminates at a nonempty perfect set. In either case, the continuum hypothesis holds for closed sets.
Suslin generalized Cantor's program to the Borel sets and the analytic sets as well. This project, however, stalled in the early twentieth century. No one could show that more complicated sets possessed the property of being either countable or equinumerous with the real line. The natural next step was to move to the projective sets, the sets of reals definable by properties that quantify over both integers and real numbers, which form a rich hierarchy of complexity. Cantor's program for proving the continuum hypothesis stalled at a relatively low level of this projective hierarchy.
Subsequent work in set theory revealed, in a deep way, that if certain large cardinals exist, certain extremely strong infinities, then Cantor's program in fact continues throughout the entire projective hierarchy. Every projectively definable set of reals would then be either countable or equinumerous with the whole real line. This result can be seen as a fulfillment, or eventual continuation, of Cantor's original idea, though it came only decades later, in the 1960s, '70s, and '80s. It vindicates his strategy of proving the continuum hypothesis by working upward through levels of complexity.
Even so, the result on projective sets does not finish the job. It does not show that every set of real numbers is either countable or equinumerous with the whole real line. The situation we are left with is this: on one hand, we have many natural examples of countable sets, the integers, the natural numbers, the rationals, the finite binary sequences, the integer polynomials, the algebraic real numbers, the computable real numbers. On the other hand, we have many natural examples of sets of size continuum, non-trivial intervals, open sets, uncountable closed sets, the power set of the natural numbers, the space of continuous functions, and so on.
What we do not have is any natural, definable example of a set of real numbers that we genuinely expect to be intermediate in size between the countable sets and the continuum. This is how matters stood for a long time after Cantor.
The question seemed hopelessly difficult to answer. Then, in 1938, the great logician Kurt Gödel made a remarkable observation: he proved that it is consistent with the axioms of set theory that the continuum hypothesis is true. He did this by giving us a model construction method. Starting from any model in which the Zermelo–Fraenkel axioms of set theory hold, Gödel described a way of constructing a certain submodel of that model, now known as the constructible universe, or L.
Gödel proved that the continuum hypothesis is true in L. He also proved that the axiom of choice is true there, so even without assuming the axiom of choice in the ambient set theory, it holds in the constructible universe. This result therefore shows simultaneously that if the axioms of set theory are consistent without the axiom of choice or the continuum hypothesis, then they are also consistent with both the axiom of choice and the continuum hypothesis being true. It is quite a remarkable achievement.
At bottom, this model construction method is very similar to the logic underlying what occurs in geometry, when one considers the parallel postulate and the question of whether it can be proved from the other axioms. That was eventually shown not to be the case by a similar kind of model construction. If one takes the Poincaré disk model of hyperbolic geometry, for example, one obtains a model that can be defined relative to Euclidean geometry in which all the other axioms of geometry hold, but the parallel postulate fails. Gödel's construction is closely analogous: starting from a model of set theory, one builds another model inside it by interpreting the relevant notions differently, and in that inner model the continuum hypothesis is true.
But does Gödel's result actually answer the continuum hypothesis question, and how is proving consistency different from proving truth? To know that the continuum hypothesis is consistent, or even relatively consistent, is simply to show that we cannot refute it. There can be no proof of the negation of the continuum hypothesis from the axioms of set theory, unless those axioms are themselves inconsistent, because Gödel showed that if the axioms are consistent, they remain consistent with the continuum hypothesis being true. What Gödel established, in other words, is that one cannot prove the continuum hypothesis is false unless one can already derive a contradiction from the axioms of set theory without any extra assumption.
This does not quite prove that the continuum hypothesis is true; it shows only that one cannot prove it is false, and that is not the same thing. There matters stood for another twenty-five years.
Gödel's result in 1938 established that the continuum hypothesis is relatively consistent with the other axioms of set theory, but it remained unknown whether it was actually provable from them. Cohen showed that it is not. He proved the analogous result for the negation of the continuum hypothesis: if the axioms of set theory are consistent, then they are also consistent with the continuum hypothesis being false.
The situation is thus precisely analogous to that of the parallel postulate in geometry. Just as the parallel postulate can neither be proved nor refuted from the other axioms of geometry, the continuum hypothesis can neither be proved nor refuted from the other axioms of set theory. If those axioms are consistent at all, then it is consistent with them that CH is true, and equally consistent with them that CH is false.
This latter result was proved by Cohen using the method of forcing, which is also a model-construction method. Whereas Gödel had started with a model of set theory and built an inner model, a smaller model contained within it, in which the axioms of set theory held together with the axiom of choice and the continuum hypothesis, Cohen's argument went in the other direction. Given a model of set theory, he constructed an outer model, a larger model, in which the continuum hypothesis fails.
That method has since been developed far beyond Cohen's original applications. We now have hundreds, perhaps thousands, of forcing arguments in set theory. To say that a statement is independent means that it can neither be proved nor refuted from the other axioms, assuming those axioms are consistent. What has emerged is that not only is the continuum hypothesis independent, but almost every non-trivial statement in infinite combinatorics and set theory has been recognized to share this same status.
We are thus confronted with a pervasive, ubiquitous independence phenomenon in set theory, with hundreds or thousands of statements each bearing exactly the same relationship to the axioms of set theory as the continuum hypothesis does.
Gödel had hoped to settle the continuum hypothesis by means of adopting extremely strong axioms of infinity. He hoped that the large cardinal axioms would resolve the continuum problem. We have different concepts of infinity going far beyond the continuum: inaccessible cardinals, Mahlo cardinals, weakly compact cardinals, Ramsey cardinals, measurable cardinals, supercompact cardinals. There is an enormous hierarchy of strength among these axioms, and they are now widely recognized as the strongest known axioms in mathematics.
Gödel's hopes of settling the continuum hypothesis on the basis of large cardinal axioms were essentially dashed by the Lévy–Solovay theorem, which shows that none of the large cardinal axioms currently known can settle the continuum hypothesis. The argument uses Cohen's method of forcing. One can show that all of the large cardinal axioms are preserved by what are called small forcing extensions, forcing with a partial order of size less than the large cardinal in question.
The relevance of this result is that one can turn the continuum hypothesis on and off by passing to a forcing extension, and that forcing is a small forcing extension. Therefore, whatever strong axioms of infinity happen to be true, one can make CH true or false, true, false, true, false, in iterated succession by forcing arguments that preserve the large cardinals. It cannot be the case that those large cardinals imply CH is true, nor that they imply CH is false, for precisely this reason.
In other words, the continuum hypothesis is independent not only of ZF and ZFC, but also of every extension of ZFC obtained by adding any of the known large cardinal axioms.
How are we to settle the CH question? Is it true or not? Embedded in that very way of asking is a certain philosophical perspective. Do we think there is a fact of the matter about whether the continuum hypothesis is true or not? This brings us to the debate on pluralism in the foundations of mathematics, and particularly the foundations of set theory.
One philosophical attitude toward the foundations of set theory holds that there is a unique set-theoretic reality for mathematics that we are trying to axiomatize with our axioms of set theory. There is a unique cumulative hierarchy of sets, achieved by starting with nothing and constructing a sequence of levels by adding all possible subsets of earlier elements. Building up the cumulative hierarchy in this manner, we arrive at the unique intended model of set theory: the set-theoretic universe. In that universe, either the continuum hypothesis is true or it is not, and the fact that ZFC, the Zermelo–Fraenkel axioms, fails to settle it, even when augmented with large cardinal axioms, tells us only about the weakness of those theories rather than about what is really the case.
This position is known as the universe view. On this view, every set-theoretic question, and indeed every mathematical question, if we interpret mathematical assertions within set theory, has a final, definitive truth value in the one intended set-theoretic universe. If you hold the universe view, then you believe there is a determinate answer to whether CH is true or false, and we are engaged in the project of discovering what that answer is, by whatever means might be available to us.
An alternative perspective is known as the multiverse view, or set-theoretic pluralism. This is the idea that there are multiple distinct concepts of set, not one set-theoretic reality, but rather multiple independent set-theoretic worlds, some in which the continuum hypothesis is true and some in which it is false. The multiverse view proceeds from the central discoveries of set theory over the past half-century, which have been precisely about constructing different models of set theory so as to exhibit different combinations of truths. We may want CH to be true, or we may want it to be false; we may want Martin's axiom to hold, or Suslin trees to exist or not exist; and what set theorists do today is build a model of set theory exhibiting exactly that desired pattern.
One can understand this situation as leading to the view that there are genuinely multiple distinct concepts of set, different coherent ways of thinking about what sets are, with each model of set theory providing a possible picture of the way set theory might be. The crucial point is that these pictures are mutually incompatible, and so from this perspective there may simply be no unique answer to the question of whether CH is true or false.
There is a certain strategy for solving the continuum problem known as the dream solution, and it is what many set theorists have hoped for. The idea is to identify a natural principle that is obviously true in the intended sense of set theory, the missing axiom, as it were, one that everyone agrees captures our core conception of the concept of set, and which also has the property of implying CH or implying not-CH. This is how the universe-minded set theorist hopes to settle the continuum problem: find the missing principle, secure universal agreement that it deserves to be an axiom, and then prove from it that the continuum hypothesis goes one way or the other.
I have argued, however, that this is a mirage, that the dream solution is impossible. The reason is that over the past half-century we have developed an enormous familiarity with a vast array of models of set theory, constructed by the forcing method and by inner model constructions, and they all seem completely set-theoretic. Some of these models have CH true and some have CH false, and we have a deep understanding of what it is like to live in worlds where the continuum hypothesis holds and what it is like to live in worlds where it fails.
The situation is therefore not merely that CH is independent and we are left in ignorance about which way it goes. Rather, we have an extremely deep experience of inhabiting these mathematical worlds with their different outcomes for the CH question. Because of that, if someone were to propose a principle that settled CH one way or the other, it would directly contradict our experience in the contrary worlds. We could never accept such a principle as manifestly true for sets, precisely because we already have experience of the contrary situations, and that experience shows those conceptions of set to be perfectly reasonable and fully set-theoretic. There is nothing wrong with those conceptions.
Let me read a passage from my book on this point. What I wrote was: "Our situation with CH is not merely that it is formally independent and we have no additional knowledge about whether it is true or not. Rather, we have an informed, deep understanding of how it could be that CH is true and how it could be that CH fails. We know how to build the CH worlds and the not-CH worlds from one another. Set theorists today grew up in these worlds, comparing them and moving from one to another while controlling other subtle features about them. Consequently, if someone were to present a new set-theoretic principle φ and prove that it implies not-CH, say, then we could no longer look upon φ as manifestly true for sets. To do so would negate our experience in the CH worlds, which we found to be perfectly set-theoretic. It would be like someone proposing a principle implying that only Brooklyn really exists, whereas we already know about Manhattan and the other boroughs."
To close, I want to draw the analogy with geometry a little more fully. Are we monist or pluralist about geometry? I think almost everyone today is a pluralist. We recognize that there are different kinds of geometry: Euclidean geometry, various non-Euclidean geometries, spherical geometry, hyperbolic space, and so on, and they are all fully real. One can be a Platonist about Euclidean geometry and simultaneously a Platonist about non-Euclidean geometry. The great geometers have developed a deep understanding of the nature of, say, hyperbolic space: what it would actually be like to move around in it. You can find videos on YouTube that illustrate the bizarre visual experiences one would have in those geometries, and the finest geometers possess extremely deep insight into the nature of these alternative spaces, insight that allows them to recognize which facts are true and to prove theorems about them.
These different geometric conceptions give rise to fully real mathematical realities that instantiate their respective perspectives. What I am arguing, in advancing the pluralist view and the set-theoretic multiverse, is that the situation in set theory is exactly the same. For thousands of years, geometry was understood to be about the one true geometry of physical space. But with the discovery of non-Euclidean geometry, that perspective splintered into a spectrum of different possible conceptions, each now taken as fully real. That is the pluralist nature of geometry.
Similarly, in set theory, there was previously the idea that the subject was about the one true set-theoretic universe. But we have come to recognize that this concept splinters, and what we are faced with is a diversity of different set-theoretic conceptions, alternative conceptions that give different answers to the Continuum Hypothesis and to many other questions, and they are all fully real concepts of set. In a sense, the multiverse picture provides an answer to the CH question: the answer is that it depends on which set-theoretic universe one inhabits. The CH holds in some universes and fails in others, and we have a deep understanding of how to pass between those universes, through the method of forcing or by passing to inner models, in just the same way that we have a deep understanding of how the parallel postulate behaves across the different geometries.
I hope you have enjoyed this discussion of the Continuum Hypothesis and the issues surrounding set-theoretic pluralism. Thank you very much.