Either there is an empty set, or there isn't. If there is one, I win. If not, then let S be the set of all empty sets. There are none, so S is the empty set and I win again.
The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism".
What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down.
If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement:
> In mathematics you don't understand things. You just get used to them.
Yep, one of my favorite math quotes and highly under-rated.
One thing I've found is that if you find yourself using the wrong "types", you're almost certainly going in the wrong direction. For example, if you're trying to solve a complex analysis exercise and you find yourself thinking about how the complex numbers = the real-coefficient polynomials modulo the ideal (x^2+1), STOP: You're getting nowhere! :)