What is ZFC and why is it important?
I googled it and it seems to be a bunch of stuff about abstract sets. As someone who did ok in math at high school, I find it hard to understand why people are so obsessed with sets. They're just collections of elements and should be very simple, right? So why do we need complicated equations to describe them?
4 replies
Pure math is build from the ground using axioms. ZFC is one axiomatization of set theory that has no contradictions, and that's why it is so important within the math community. It basically allows you to build all of set theory, and if you include the axiom of choice, it also has a applications on different fields of math like analysis.
If you haven't, check out Russell's paradox.
Does ZFC have the set of all sets that don't contain themselves, or is there a rule against that one?
Such set cannot exists under ZFC. Or rather, you cannot construct the set used in Russell's paradox using ZFC.
The more abstract and general a theory is, the more different things it can be applied to. If you can figure out how to describe any problem in terms of sets, you immediately have all the theorems of ZFC at your disposal.