Set Theory

Abstract

0009 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,OX) consisting of a topological space X and a sheaf of rings OX on it. A topological space is a pair (X, τ) consisting of a set X and a set of subsets τ ⊂ P(X) satisfying the axioms of a topology. And so on and so forth.