> If you don't know why set theory is important, it is because set theory is the foundation of all of mathematics.

Sorry to burst your bubble, but as far as we know, that isn't true in the slightest. It's a logical positivist view abandoned after Goedel and Turing.

At best: What we hope is true is that there often is some axiomatic system where a specific mathematical lemma makes sense when redefined into something similar but not the same.