I understand how integers and rationals are expressed/derived in ZFC. But what about the irrational numbers? Can they also be expressed? If not, are there other axiomatic set theories able to express them? As for Dedekind cuts, from my understanding (maybe wrong), any irrational in question must have a 1-1 explicit function to a natural number, in order for the method to work (As an example the square root of 2 has such a function).
It boils down to this: is there an isomorphism between the the set of irrationals (as "just" numbers) and a a set of pure sets?
Dedekind cuts are the most common way. The idea is that, given the rationals, you take the reals to be all the downward closed sets of the rationals (including all the ones with no greatest element). For a concrete example, $\sqrt{2}$ is $\{x\in\mathbb{Q}:x^2<2\}$.
The version I'm chiefly familiar with is Quine's from "Set Theory and its Logic". He defines a rational $a/b$ as $\{x+(x+y)^2:x\cdot b < a\cdot y\}\subset\mathbb{N}$. Then given a collection of such rationals $A$, their upper bound is $\bigcup A$. Most of these upper bounds fail to be other rationals themselves. In this treatment, rationals are a subset of the reals as they are intuitively, and "$\leq$" reduces to $\subseteq$.