The Future of Modern Set Theory

Annals of the Japan Association for Philosophy of Science, March 1994 1

James E. BAUMGARTNER

The biggest obstacle to discussing the future of set theory is surely the fact that set theorists are almost universally pessimistic. There appears to be a powerful feeling that the end of set theory is nigh, and as far as I can tell this seems to have been true for about seventy years, almost as long as there have been set theorists. Perhaps to be a set theorist is to be pessimistic. Of course, there are other fields that suffer from similar attitudes, physics, for example. Consider the article by John Maddox in the November 25 issue of Nature entitled "Has physics come to an end ?" Now many would think that set theory is a great deal more secure than physics. Nearly all mathematics can be fit inside it, for example, and there are proofs of the incompleteness and undecidability of even tiny portions of set theory. But that is not the point, the pessimists say. The question is not whether there remain undecidable propositions in set theory, but whether there are interesting, fundamental, essentially set-theoretical issues still to be decided. Number theory can certainly be fit into set theory, but it is unfair to appropriate hard number- theoretical questions and pass them off as set theory.

A good view of the situation may be obtained by considering briefly the history of set theory. It was only a little more than a hundred years ago that Cantor began the study of cardinal and ordinal numbers that gave rise to the subject. And in some sense he settled most of the simple questions that came to mind. An exception was the Continuum Hypothesis, of course; Hilbert was kind enough to give it a prominent position in his problem list, but most people expected it to have a straightforward although possibly difficult solution. And once it was settled, what would remain of set theory ? Another result that postdated Cantor was the result of J.Konig about cardinal exponentiation. Nowadays it is common to describe that result as simply saying that cf(2ƒÉ)>ƒÉ for any cardinal .1. Until rather recently, however, it was common to see Konig's result stated in a fairly elaborate form after all, it said everything about cardinal exponentiation that

Cantor had neglected to find, and there surely was not so much to the subject.

Cantor's work on cardinal and ordinal numbers was finally applied in the 1920's by Hausdorff and the Polish set theorists, particularly including Sierpinski. A variety of topological spaces with interesting properties were constructed, as were more general set-theoretical objects like families of almost-disjoint sets, ordered sets

Dartmouth College, Department of Mathematics and Computer Science, Hanover, New Hampshire 03755 USA

-187- 2 James E. BAUMGARTNER vol. 8 and the like. And only the known properties of cardinal and ordinal numbers were used. But was this set theory ? Or was it topology (in the case of Hausdorff) or infinite combinatorics (in the case of Sierpinski)? In his book Hypothese du Continu Sierpinski even worked out dozens of propositions equivalent to the Continuum Hypothesis. Surely once that was settled nothing else would be left. The most significant development occurred only in 1939 when Godel defined the constructible sets and showed that the class L of such sets is a natural inner model for the axioms ZFC of set theory, and that in addition the Generalized Continuum Hypothesis (GCH) is true in L. So the Continuum Hypothesis is at least consistent with the other axioms of set theory. Of course, this sheds little light on the actual truth or falsity of the Continuum Hypothesis, and the real value of Godel's result was not well understood by those working in the field at the time. The result must have seemed isolated, with little clear application to the real problems of set theory. The following two decades saw the birth in Hungary of the partition calculus in work of Erdos, Rado and others. This work made free use of GCH since by Godel's result GCH could not be disproved from the axioms; a joke was even made that ZF+GCH should be known as ZFE (with E standing for Erdos). Nonetheless a substantial body of results was obtained from ZFC alone, but the real character of the work was not fundamental set theory, but rather more like the infinite com binatorics of Sierpinski. If a paper with the title of this one had been written at the end of the 1950's, it would undoubtedly have been pessimistic. With the exception of Godel's result, which had had no successors, set theory was in much the same condition it had been left in by Cantor. True, considerable application of set-theoretical ideas had appeared in the work of Hausdorff, Sierpinski, Erdos, Rado, and others, but not much progress on the fundamental questions had been made in 75 years. What could be the future of such a subject ?

Then came the 1960's. Unexpectedly an enormous variety of fundamental results began to appear. One of the first was an observation of Scott concerning an old definition of Ulam. Ulam considered infinite sets which bear o-complete {0, 1}- measure defined on all subsets. From the definitions was derived the modern idea of a measurable cardinal, and Scott proved that if there is a measurable cardinal then V •‚ L. Of course, one can view this as saying that in L there are no measurable cardinals, but this is complicated by the fact that if there is a measurable cardinal then there is a model for ZFC in which necessarily not all sets are constructible. This is quite different from the situation for GCH. And this was almost immediately followed by Cohen's famous work on forcing and generic sets, a fundamental, new construction that finally established the consistency of the negation of the Continuum Hypothesis. Moreover, it was clear almost from the beginning that Cohen's construction, especially as modified by Scott

-188- No. 4 The Future of Modern Set Theory 3 and Solovay, was extremely powerful and could be used to settle a vast number of old problems that had been eclipsed by the concern for the Continuum Hypothesis. Cohen was also able to settle questions about the provability of various forms of the Axiom of Choice (usually they were not provable). Suddenly set theory was a hot topic, and many bright young graduate students were attracted by it. In 1967 there was a six-week meeting at UCLA including nearly all those working in set theory. The vast majority of the participants were under 30. If a paper about the future of set theory had been written at that time, and several were, for the first time the tone would have been optimistic. The explosion in set theory had some powerful reverberations. There were a great number of results and independence methods developed very quickly, and Solovay even found ways to force over models containing large cardinals, inacces sible or measurable, to answer old questions. In his dissertation, Silver applied some of the technology on partition relations developed by Hajnal and others to make much clearer the relation between a universe containing a large cardinal (such as an Erdos cardinal) and L. And finally, by the end of the decade Jensen had made tremendous progress into the theory of L itself, including the definition of various combinatorial properties that had far-reaching consequences in many different areas of set theory and model theory.

The work done in the decade of the 1960's must have been largely unimaginable only a few years earlier. For one thing, even the character of set theory seemed to have changed. What objects does set theory study ? The work of Cohen, Jensen, and their successors suggested strongly that it is not the properties of sets so much as the properties of models of set theory. The method of forcing really provided a way to convert one model into another. Jensen's work showed the importance of considering mappings between various partial models of set theory that were either elementary or nearly so (say, ƒ°n for some n), and this effort was extended another way by Kunen's dissertation in which an inner model like L for the existence of a measurable cardinal was studied in connection with elementary mappings obtained from iteration of the natural elementary embedding obtained from the measure.

This now provides a way of placing set theory within the usual categyry theoretic view of mathematics. If every part of mathematics is distinguished by the so-called objects and morphisms that it studies, then we can describe the objects of set theory as models of (part of) set theory, and the morphisms as mappings that are elementary (or nearly so). Now, this characterization is not perfect but it is good enough to suggest why set theory has taken its place as a clearly distinguishable part of mathematics. Most of the recent work in set theory has been in the areas established during the 1960's. Beginning with the work of Mitchell on models with many measurable

-189- 4 James E. BAUMGARTNER Vol. 8 cardinals, a great deal of work has been done studying inner models for various species of large cardinals, and much remains to be done. Even the kinds of large cardinals studied have been suggested by the considerations of inner model theory. The core model K, a kind of analogue of L of interest in many models of large cardinals, has been developed by Jensen and Dodd. Woodin and others have investigated L[R], the collection of all sets constructible from the reals, a model with remarkable stability properties. The huge expansion of the theory of forcing and generic sets has continued, giving rise to deep results about forcing methods, especially iterated forcing, and leading to a very extensive study of the iceberg of null sets and sets with the property of Baire, only the tip of which was imagined by Hausdorff. And there have been developments outside these areas as well. For example, descriptive set theory, the study of definability properties of the reals together with the Axiom of Determinateness (AD), reached a climax not long ago with the proof by Martin and Steel that AD is consistent with ZF. The proof used forcing and large cardidnal, and the development of definability theory has reached into the rest of mathematics via the study of Borel structures. In recent work, Shelah has obtained tremendous results on powers of singular cardinals in ZFC alone, sufficient to show the folly of the one-time belief that Konig's Theorem said all there was to say. The ultimate dimensions of this investi gation are still little understood, and very little work has been done applying the theories of forcing and large cardinals to demonstrate its limits. So what remains to be done in set theory ? In many areas that question would be regarded as a feeble joke. For decades, set theory has lived with a sense of impending closure that simply does not exist any more. There is a great deal of work cut out for a potential pessimist. Perhaps the most pessimistic view is the following: For many years, set theory was a subject that required only about a semester of intense work to master. Now it is maturing as a mathematical disci pline, and prospective students are faced with several years of study to reach the research frontier. The future of the subject, of course, rests with the young people. But all this is true of many of the most interesting harts of mathematics. What is the future of set theory ? In view of its past, it would surely be foolish to try to cast its future in terms of the solution of a short list of specific problems. All we can say is that it will have a future, and that we can expect many new and interesting problems to drive it.

-190-