>>13
set theory is a nice and simple thing that you can use to rigorously define other things, but nothing is really stopping you from trying to define everything directly, or from using an alternative vocabulary to define your constructs. It can aid your intuition to think about things from this point of view. The details in the set theoretic definition can become very arbitrary and distract you from what it is all really supposed to represent. You do need some underlying logical framework though, if you want to be able to prove anything. But the choices used in setting up that framework are arbitrary.