Sets, Logic, and Categories

Abstract

The rich multiplicity of Mathematical objects and the proofs of theorems about them can be set out formally with absolute precision on a remarkably parsimonious base. Thus almost all the objects of Mathematics can be described as sets: A natural number is a set of sets (a cardinal), a rational number is a set of pairs (an equivalence class), a real number is a set of rationals (a Dedekind cut), and a function is a set of ordered pairs (a table of values). Similarly the theorems of Mathematics can all be written as formulas in a very parsimonious formal language which uses only set-membership, the basic connectives of logic (or, not, there exists) and the needed primitive terms of each subject (thus “point” and “line” for incidence geometry). Finally, most of the proofs of Mathematical theorems can be stated with absolute rigor as a sequence of inferences, each an instance of a finite number of basic schemes of inference.