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Homotopy Type Theory has been receiving a lot of hype during the last couple of years and been hailed as the most effective foundation of all Mathematics compared with Set Theory.

My question: If HOTT is the foundation of all mathematics, will that pronounce death of Set Theory research ? or else the Set Theory is different descipline in which research is conducted independent of whether it is or isn't the foundation of Mathematics ? What sort of topics do set theorist research on ?

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    $\begingroup$ Easy. Interpret HTT in $\sf ZFC$+sufficiently many inaccessible cardinals. Proceed with $\sf ZFC$. In either case, you can't quite avoid independence claims when you start fiddling with uncountably infinite objects. Set theory is our tool to understand infinite objects better. Since uncountable objects will not disappear any time soon, set theory is not going anywhere. $\endgroup$ – Asaf Karagila May 20 '15 at 16:29
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    $\begingroup$ Also, can you phrase this question in a manner which is not speculative? It seems that you're interested in what sort of research set theorists do. And that's a good question, which has been answered thoroughly before. But the question as written is somewhat insulting to read, as a set theorist (which I am sure is not your intention). $\endgroup$ – Asaf Karagila May 20 '15 at 16:32
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    $\begingroup$ @AsafKaragila insulted because it threatens your livley-hood, or insulted because Op makes bold claim with 0 knowledge of ST/HTT? (Should this question be taken seriously?) $\endgroup$ – Zach466920 May 20 '15 at 16:45
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    $\begingroup$ @Zach466920: The latter; and the first part of your comment too. $\endgroup$ – Asaf Karagila May 20 '15 at 16:48
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    $\begingroup$ @Asaf can correct me if I'm wrong, but stuff focused on foundational questions is but a minor part of set theory; I see no reason questions about cardinal strength will stop being interesting given some other foundation of mathematics. $\endgroup$ – user98602 May 20 '15 at 17:27
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Others have addressed the concrete question of "what do set theorists do?", so let me take a stab at the speculative "If HoTT is the foundation of all mathematics, will that pronounce death of Set Theory research?" I can imagine many possible futures in the foundations of mathematics, such as:

  • Set theory remains ascendant. In this case, there would probably not be much change in set theory research.
  • Some other foundation, such as HoTT, becomes dominant in the same way that set theory is now. This would take a long time to happen, but it's at least conceivable. In this case, set theory would be somewhat reduced in foundational importance, but set theory research as an independent subject would, I think, be largely unaffected. From a HoTT point of view, the set theory that "set theorists" do could be called "the study of classical well-foundedness", and it is an interesting subject regardless of its foundational importance or unimportance. Moreover, formulating this theory within another theory like HoTT might open up new ways to look at it and new directions of research.
  • Mathematics is freed from dependence on a single foundation. To some people this is the most attractive possibility, and it also seems more likely than the second possibility, at least in the short to medium term (it's hard to imagine set theorists, at least, giving up the view of set theory as foundational!). In this case, set theorists would probably continue to consider set theory as foundational, HoTT theorists would consider HoTT as foundational, and likewise for people working on any other foundational systems, while mathematicians not working in a subject closely related to foundations would probably remain mostly as indifferent to foundations as they are today. Set theory research would probably be almost entirely unaffected, except for the same possibility of new connections and directions mentioned above.

Of course, this is entirely speculation, and as we all know it is difficult to make predictions, especially about the future. Others may speculate differently than I.

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It seems that for you "foundational" means concerning the foundation of ordinary (non-set theoretic) mathematics. In this sense, I would claim that this sort of foundational concern has been a very minor aspect of set theory for decades. Moreover, set theory has never served as foundation for mathematics in any deep practical sense. Probably, Homotopy Type Theory will not really serve as foundation for mathematics in any practical sense either.

This really is not due to any particular nature of set theory or any other foundations. The majority of mathematicians do not care about foundations. They care about the ideas and objects they are working with not how in particular these objects are formalized. It quite conceivable that in the future other foundational ideas may appear as a language tool. But most likely, mathematician will just casually throw arount these new terms from their new foundations, just like mathematicians now casually throw out the word "sets", "infinity", "cardinality", "intersection", "functions", etc.

As for my claim that foundation is a very minor concern for most set theorist: I do think there are many set theorist that are currently working on formalizing any branch of natural mathematics in set theory. I think set theory has long evolved from the study of sets in your sense. Set theory is now the study of the combinatorics of infinity and certain logical phenomena. Homotopy type theory or something else may replace set theory as the prominent language for doing ordinary math, but there is nothing currently available that appear to be able to replace this study of infinity.

Probably, this is because set theory and infinity are inseparable. Since ancient times people were vaguely familiar with infinity. The beginning of the systematic study of infinity is one and the same as the birth of set theory. Cantor was probably the first person to mathematically study different forms of infinity and this was precisely when set theory began. Calculus existed independently of set theory. Did the mathematical branch of infinity really even exists before Cantor began set theory?

On the logical side, modern set theory attempts to produce independence and consistency results. No other foundation has quite as many tools (such as forcing and inner models in set theory) to be able to produce these types of results. If you are interested in the philosophical aspect of some theorem and you know it can be formalizes in several foundations, for practical purpose you would probably use the available tools of set theory to analyze it logical content.

On large cardinals, they are indeed used as foundations but not in your sense. They are foundations for descriptive set theory and infinitary combinatorics. In the current state of mathematics, infinity and set theory is the same thing, so I do not think you would consider this aspect as foundation.

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A few points, in no particular order

  • In mathematics, there are no decrees. Nobody has the right or power to decree any particular formal system as "the" foundations. You do things your own way, period. Foundational pluralism is, in some sense, inevitable.

  • We should distinguish between "set theory" and "material set theory." Your real question is probably "what is the future of material set theory"?

  • Sets can be defined in HoTT as homotopy $0$-types. Ergo, HoTT encompasses set theory. Its not an alternative to set theory, its an enlargement. Furthermore, sets continue playing a reasonably central position in HoTT.

  • HoTT is completely different to material set theory, but I'm not convinced its completely different to set theory. Homotopy types are a lot like sets.

  • Most theorems of ZFC, except those that deal with the cumulative hierarchy in a rather direct way, should have analogues in HoTT. Most of that knowledge should "translate over."

  • Its possible to interpret HoTT in ZFC; ergo, ZFC can still be regarded as being "the" foundations by anyone who prefers this perspective.

  • Arguably, taking emphasis away from the cumulative hierarchy (which type-theoretic foundations tend to do) is good idea. But this is a very controversial position; set theorists live and breathe the cumulative hierarchy.

  • You can get away from the cumulative hierarchy without going all the way to type theory; see also, ETCS.

  • Topos theory is a useful generalization of set theory with applications to not only set theory, but also computability theory and geometry. There's such a thing as "topos-theoretic forcing."

  • Its not currently known whether or not HoTT has anything interesting to say about the structure of the category of sets that cannot be straightforwardly investigated within a ZFC-like framework.

I have personally found it rather difficult to learn type theory; one reason for this is that the whole time I'm thinking: "Isn't this just set theory over constructive logic minus the cumulative hierarchy?" The same can be said of HoTT with "set theory" replaced by "$\infty$-groupoid theory." Until I understand the difference better, I'm not qualified to comment any further.

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Shelah has written an article on the future of set theory. Here's a (somewhat outdated) write-up of his on the sort of problems he was interested in late 1990s.

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