TITLE: What is the future of Set Theory if it is NOT the foundation of Mathematics?
QUESTION [8 upvotes]: 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 ?

REPLY [8 votes]: 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.