What is the Relevancy of Other Foundations of Mathematics Besides ZFC

Question

In mathematics, ZFC set theory has made a significant splash; it has lead the way for very important concepts such as the incompleteness theorem, the continuum hypothesis, aleph numbers, and innumerably many other things. However, other foundations of math, such as HoTT and category theory, haven't brought forth such significant results that I know of (save applications in computer science). Has HoTT, Category Theory, or any other foundation besides ZFC yeilded significant results or have the potential for significant results? Thanks in advance!

Answer 1

First of all, to correct a common misconception: category theory is not a foundation for mathematics. Category theory is a language that can be used to describe lots of kinds of mathematics, including some foundations of mathematics, such as Lawvere's Elementary Theory of the Category of Sets; but those foundations are not the same as category theory.

Second, I would say that one reason ZFC gave rise to so many important consequences is that it was the first precise foundation for mathematics. In some sense, this is a historical accident; in principle another foundation such as ETCS or type theory could have been developed first, and in that case it would have been that foundation giving rise to all those significant developments, most of which are largely insensitive to the particular foundation of mathematics in which they are formulated.

Now that we are familiar with the notion of "foundation of mathematics", and many of the benefits of having such a foundation have already been reaped, we shouldn't expect "new" foundations to impact mathematics in the same way that ZFC did. Instead, their effects will be more incremental: providing better ways to formalize certain parts of mathematics, or enabling new mathematics that wasn't possible in ZFC. But this does not mean that, taken in their own, they are any less "foundational" than ZFC.

Answer 2

A meta comment: Zermelo set theory was initially developed over a century ago. HoTT has only been around a decade or so. Gödel's incompleteness theorems were published in 1931, over two decades after Zermelo set theory was initially developed. It seems a bit unfair to expect that HoTT should have already produced similarly groundbreaking new results.