The Stacks project

3.2 Everything is a set

Most mathematicians think of set theory as providing the basic foundations for mathematics. So how does this really work? For example, how do we translate the sentence “$X$ is a scheme” into set theory? Well, we just unravel the definitions: A scheme is a locally ringed space such that every point has an open neighbourhood which is an affine scheme. A locally ringed space is a ringed space such that every stalk of the structure sheaf is a local ring. A ringed space is a pair $(X, \mathcal{O}_ X)$ consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}_ X$ on it. A topological space is a pair $(X, \tau )$ consisting of a set $X$ and a set of subsets $\tau \subset \mathcal{P}(X)$ satisfying the axioms of a topology. And so on and so forth.

So how, given a set $S$ would we recognize whether it is a scheme? The first thing we look for is whether the set $S$ is an ordered pair. This is defined (see [Jech], page 7) as saying that $S$ has the form $(a, b) := \{ \{ a\} , \{ a, b\} \} $ for some sets $a, b$. If this is the case, then we would take a look to see whether $a$ is an ordered pair $(c, d)$. If so we would check whether $d \subset \mathcal{P}(c)$, and if so whether $d$ forms the collection of sets for a topology on the set $c$. And so on and so forth.

So even though it would take a considerable amount of work to write a complete formula $\phi _{scheme}(x)$ with one free variable $x$ in set theory that expresses the notion “$x$ is a scheme”, it is possible to do so. The same thing should be true for any mathematical object.