What is the intuition for why all of math can be developed using set theory? I learnt that the formal language of pure set theory allows one to formalize all mathematical notions and arguments. The language just has one non logical symbol(!) "the belongs to relation", this is very surprising to me, is there any intuition for why this is true? I mean its surprising that math has such a simple "building block".
Please note that I'm asking why(as in why is it intuitive) not how [I know how]
 A: In general most areas of math you can describe the objects of interest as ordered $n$-tuples where the first element of the tuple will be a set and the rest of the elements will be operations, relations, etc. For example a group is defined to be a pair $(G,*)$ where $*$ is an operation satisfying some axioms. Or $(X,d)$ is a said to be a metric space if $d: X\times X \rightarrow \mathbb{R}_{\leq 0}$ satisfying symmetry, triangle equality, and point separability. The list goes on but since I can define most objects in mathematics in this way that means that one should be able to talk about most math just by being able to do the basic set operations (union, intersection, cartesian product, powerset ...) and have the basic numerical sets ($\mathbb{N}$,$\mathbb{Z}$,$\mathbb{Q}$,$\mathbb{R}$,$\mathbb{C}$). But all numerical sets can be defined using the basic set operations starting from $\mathbb{N}$. The definition of numerical sets from "simpler ones" was a project done way before the introduction of axiomatic set theory. We have that that in the early 19th century Hamilton formalizes the complex numbers starting from the reals. In the second half of the 19th century you see various attempts to formalized the set of real numbers. The two most famous constructions are the as equivalence classes of Cauchy sequences of the rationals that was done by Cantor and the second was as the set of Dedekind cuts. If you observe these constructions you notice you are using simple set theory operations to define them. Construction of the rationals from the integers is a case of a more general notion of the field of fractions of an integral domain which is itself a special case of localization. Finally the integers are built from the natural numbers through group completion. You can observe that all these constructions involve taking the cartesian product and defining equivalence classes.
One finally arrives to the natural numbers which can be defined as a class in $ZFC$ without infinity. The way these are defined are as the Von Neumann ordinals whose elements are either $0$ or a successor and are themselves either $0$ or a successor. One must assume the axiom of infinity to assure that the naturals exist as a set and thus one can apply set operations on them. Thus $ZFC$ or even $ZC$ is sufficient to do most of these constructions. Some of  more recent math such as the study of large cardinals or category theory often talk about objects that are a "too big".
