What parts of set theory are only used in set theory?

1 Answer
Dec 22, 2017

Hmmm...

Explanation:

As far as I know, all essential properties of ZF, ZFC or NBG set theory are used in some other branch of mathematics. For example, on first encounter you might think that the axiom of choice is a little esoteric and would not be used outside of set theory, but it is crucial to solving many problems in other areas.

One area of set theory that does seem more restricted to set theory is the study of large cardinals - essentially adding axioms to set theory that cardinals with particular properties exist. I know of no application of these large cardinals outside of set theory.

I spoke too soon. Even large cardinals have various applications outside of set theory - in graph theory and measure theory to name just a couple. It seems that they can simplify some proofs in category theory too.

Basically set theory provides a rich foundation useful for all kinds of mathematics.