What are the issues in modern set theory?

This is spurred by the comments to my answer here. I'm unfamiliar with set theory beyond Cohen's proof of the independence of the continuum hypothesis from ZFC. In particular, I haven't witnessed any real interaction between set-theoretic issues and the more conventional math I've studied, the sort of place where you realize "in order to really understand this problem in homotopy theory, I need to read about large cardinals." I've even gotten the feeling from several professional mathematicians I've talked to that set theory is no longer relevant, and that if someone were to find some set-theoretic flaw in their axioms (a non-standard model or somesuch), they would just ignore it and try again with different axioms.

I also don't personally care for the abstraction of set theory, but this is a bad reason to judge anything, especially at this early stage in my life, and I feel like I'd be more interested if I knew of some ways it interacted with the rest of the mathematical world. So:

• What do set theorists today care about?
• How does set theory interact with the rest of mathematics?
• (more subjective but) Would mathematicians working outside of set theory benefit from thinking about large cardinals, non-standard models, or their ilk?
• Could you recommend any books or papers that might convince a non-set theorist that the subject as it's currently practiced is worth studying?

Thanks a lot!

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Clearly, we need JDH to weigh in on this. – Arturo Magidin Mar 7 '11 at 5:50

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Arturo, you are very kind. – JDH Mar 7 '11 at 17:23

5 Answers

Set theory today is a vibrant, active research area, characterized by intense fundamental work both on set theory's own questions, arising from a deep historical wellspring of ideas, and also on the interaction of those ideas with other mathematical subjects. It is fascinating and I would encourage anyone to learn more about it.

Since the field is simply too vast to summarize easily, allow me merely to describe a few of the major topics that are actively studied in set theory today.

Large cardinals. These are the strong axioms of infinity, first studied by Cantor, which often generalize properties true of $\omega$ to a larger context, while providing a robust hierarchy of axioms increasing in consistency strength. Large cardinal axioms often express combinatorial aspects of infinity, which have powerful consequences, even low down. To give one deep example, if there are sufficiently many Woodin cardinals, then all projective sets of reals are Lebesgue measurable, a shocking but very welcome situation. You may recognize some of the various large cardinal concepts---inaccessible, Mahlo, weakly compact, indescribable, totally indescribable, unfoldable, Ramsey, measurable, tall, strong, strongly compact, supercompact, almost huge, huge and so on---and new large cardinal concepts are often introduced for a particular purpose. (For example, in recent work Thomas Johnstone and I proved that a certain forcing axiom was exactly equiconsistent with what we called the uplifting cardinals.) I encourage you to follow the Wikipedia link for more information.

Forcing. The subject of set theory came to maturity with the development of forcing, an extremely flexible technique for constructing new models of set theory from existing models. If one has a model of set theory $M$, one can construct a forcing extension $M[G]$ by adding a new ideal element $G$, which will be an $M$-generic filter for a forcing notion $\mathbb{P}$ in $M$, akin to a field extension in the sense that every object in $M[G]$ is constructible algebraically from $G$ and objects in $M$. The interaction of a model of set theory with its forcing extensions provides an extremely rich, intensely studied mathematical context.

Independence Phenomenon. The initial uses of forcing were focused on proving diverse independence results, which show that a statement of set theory is neither provable nor refutable from the basic ZFC axioms. For example, the Continuum Hypothesis is famously independent of ZFC, but we now have thousands of examples. Although it is now the norm for statements of infinite combinatorics to be independent, the phenomenon is particularly interesting when it is shown that a statement from outside set theory is independent, and there are many prominent examples.

Forcing Axioms. The first forcing axioms were often viewed as unifying combinatorial assertions that could be proved consistent by forcing and then applied by researchers with less knowledge of forcing. Thus, they tended to unify much of the power of forcing in a way that was easily employed outside the field. For example, one sees applications of Martin's Axiom undertaken by topologists or algebraists. Within set theory, however, these axioms are a focal point, viewed as expressing particularly robust collections of consequences, and there is intense work on various axioms and finding their large cardinal strength.

Inner model theory. This is a huge on-going effort to construct and understand the canonical fine-structural inner models that may exist for large cardinals, the analogues of Gödel's constructible universe $L$, but which may accommodate large cardinals. Understanding these inner models amounts in a sense to the ability to take the large cardinal concept completely apart and then fit it together again. These models have often provided a powerful tool for showing that other mathematical statements have large cardinal strength.

Cardinal characteristics of the continuum. This subject is concerned with the diverse cardinal characteristics of the continuum, such as the size of the smallest non-Lebesgue measurable set, the additivity of the null ideal or the cofinality of the order $\omega^\omega$ under eventual domination, and many others. These cardinals are all equal to the continuum under CH, but separate into a rich hierarchy of distinct notions when CH fails.

Descriptive set theory. This is the study of various complexity hierarchies at the level of the reals and sets of reals.

Borel equivalence relation theory. Arising from descriptive set theory, this subject is an exciting comparatively recent development in set theory, which provides a precise way to understand what otherwise might be a merely informal understanding of the comparative difficulty of classification problems in mathematics. The idea is that many classification problems arising in algebra, analysis or topology turn out naturally to correspond to equivalence relations on a standard Borel space. These relations fit into a natural hierarchy under the notion of Borel reducibility, and this notion provides us with a way to say that one classification problem in mathematics is at least as hard as or strictly harder than another. Researchers in this area are deeply knowledgable both about set theory and also about the subject area in which their equivalence relations arise.

Philosophy of set theory. Lastly, let me also mention the emerging subject known as the philosophy of set theory, which is concerned with some of the philosophical issues arising in set theoretic research, particularly in the context of large cardinals, such as: How can we decide when or whether to adopt new mathematical axioms? What does it mean to say that a mathematical statement is true? In what sense is there an intended model of the axioms of set theory? Much of the discussion in this area weaves together profoundly philosophical concerns with extremely technical mathematics concerning deep features of forcing, large cardinals and inner model theory.

Remark. I see in your answer to the linked question you mentioned that you may not have been exposed to much set theory at Harvard, and I find this a pity. I would encourage you to look beyond any limiting perspectives you may have encountered, and you will discover the rich, fascinating subject of set theory. The standard introductory level graduate texts would be Jech's book Set Theory and Kanamori's book The Higher Infinite, on large cardinals, and both of these are outstanding.

I apologize for this too-long answer...

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Great answer! I would also refer to Jech's Set Theory The Millennium edition preface, as well Kanamori's historical overview in The Higher Infinite. – Asaf Karagila Mar 7 '11 at 17:29

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@JDH: No apologies necessary; exactly what I was hoping for. – Arturo Magidin Mar 7 '11 at 18:30

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Apology not accepted! Great answer. – The Chaz 2.0 Mar 11 '11 at 2:39

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– Charles Stewart Jun 28 '11 at 8:36

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@Charles, of course that is excellent work, with some excellent people working on it, and it clearly has deep connections with set theory. But at the same time, my impression is that the theory arises principally from a category-theoretic perspective rather than a set-theoretic one, no? But surely set theory is a big tent. – JDH Jun 28 '11 at 11:50
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Maybe start with looking at the chapters for the Handbook of Set Theory here. As a topologist, I can say that set theory is still very useful in General Topology, as people run into many questions that are independent of set theory there, or that are helped by techniques from Set theory. Harvey Friedmann has many (I think) interesting ideas about the role of large cardinals in "normal", combinatoric questions. Shelah is probably the most prolific author in Set Theory and has quite a breadth of subjects.

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One of the professors in my department was talking to me about it the other day, he joked that logic and set theory to "conventional" mathematics is like mathematics to physics.

First of all, to me conventional mathematics is to take an idea and derive more ideas from it by logical inference - this relates to what my Algebra I professor said in the very first math lecture I attended to: mathematics is the science of deducing A from B and C.

Secondly, the aforementioned professor was joking but it was part true in a way. Set theory trickles into model theory and topology which in turn trickle into algebra and analysis. Those fields are conventional mathematics, I think.

Lastly, I can't speak completely about set theorists but I can say what I see from the very narrow point of view I have right now - the dominance of ZF was established and now there is a search for "measures of consistency" how much more do you need to assume. The axiom of choice, its negation, existence of some large cardinals, and so forth and so on. This is my narrow point of view, as someone who's mostly studying forcing and large cardinals for the past few months and I might as well be talking trash.

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Is ZF's dominance established?  It seems like it's nice to have some notion of "class," for category theory if nothing else. – Paul VanKoughnett Mar 4 '11 at 2:46
@Paul: ZF is very good in the sense that you don't "feel" you're working with the axioms, they are all very natural and describe what we thought are sets to begin with. There are ways to deal with classes, and there is the NBG axiomatic set theory which is a conservative extension of ZF, which essentially means you can prove the same things regarding to sets. NBG has classes. Still, you don't see many people work with it. – Asaf Karagila Mar 4 '11 at 5:14

One place where you need to care about set-theoretical issues is in category theory. Even once you get past the obvious 'problems', like the existence of the category of sheaves on a non-small category (one of the usual solutions is to assume at least one universe - a set that acts like $V_\kappa$ for some inaccessible $\kappa$, and work just with sets of size bounded by the size of elements of $V_\kappa$, or perhaps enough such sets such that every set in contained in some universe), there are interesting questions that are affected by set-theory like Vopěnka's principle - see the nLab page for a brief discussion.

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Two of the set theorists recently working and collaborating with algebrists on category theoretic issues are Joan Bagaria, bcnsets.ub.es/researchers/Joan.Bagaria and Adrian Mathias. See for example arxiv.org/abs/1101.2792 "Definable orthogonality classes in accessible categories are small", by Bagaria, Mathias, Carles Casacuberta, and Jiri Rosicky. – Andres Caicedo Mar 7 '11 at 6:54

I'll take a stab at this part:

"How does set theory interact with the rest of mathematics?"

Henno Brandsma mentioned that set theory is useful in topology. Another area, which has quite a few analogies to topology, is measure theory, and by extension, probability theory. We are very interested in, for example, infinite product spaces, because many useful probabilistic models have infinitely many random variables. But we can't just summon them into existence without some care. To do so in first-order logic requires the axiom of choice (or something like it). But then non-measurable sets exist, so every set of outcomes must be accompanied by a $\sigma$-algebra of sets for which measurement (probability or volume) makes sense...

Further, much like in topology, our basic definitions rely quite heavily on "set-construction" axioms like comprehension, union and powerset. For example, the common operation "take the coarsest $\sigma$-algebra $\mathcal{A}$ such that $\phi(\mathcal{A})$" appeals directly to those axioms: powerset twice, then comprehension, and then union (with one indirection, from an intersection of uncountably many $\sigma$-algebras).

So probability theory is intricately tied with the axiom of choice, in such a way that anybody doing serious work in it is always aware of it. And we use the axioms directly, even if we often forget that that's what we're doing.

"I've even gotten the feeling from several professional mathematicians I've talked to that set theory is no longer relevant, and that if someone were to find some set-theoretic flaw in their axioms (a non-standard model or somesuch), they would just ignore it and try again with different axioms."

What else would we do but try again? :)

The real question is: how easy would it be? Much of analysis would be fine; the reals seem to have the same properties no matter what theory you build them in. Measure theory and probability theory, however, which are so tied to set theory, would (I think) come crashing down - everything would be suspect, and would have to be rebuilt almost from scratch. What a terrifying and exciting idea!

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Actually, when you don't assume the axiom of choice the reals can be a countable union of countable sets, and have Baire first category properties. – Asaf Karagila Mar 7 '11 at 6:52
@Asaf: That's really fascinating! When I wrote "have the same properties" I was referring to the field axioms, arithmetic properties of limits, and such. But I wrote "seems to" in case there was some construction that would give them subtly different properties from the typical reals - like the one you brought up. :) – Neil Toronto Mar 7 '11 at 16:34
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