Groucho Marxism

Most people assume that numbers are the basic building blocks of mathematics. But mathematicians take sets to be the basic building blocks and construct numbers from them. In this blog post I will explain how this is done. First we need to construct the natural numbers: 0, 1, 2, 3, and so on. There are several ways to do this; the standard method, due to the Hungarian mathematician Jon von Neumann, is to define 0 := {} (the empty set) then define a+1 := a∪{a} for each natural number a. Next we need to construct the integers: the natural numbers plus -1,-2, -3, and so on. This is done using ordered pairs (a,b) of natural numbers. Again there are several ways to define an ordered pair; the standard method, due to the Polish mathematician Kazimierz Kuratowski, is to define (a,b) := {{a},{a,b}}.

We say two ordered pairs of natural numbers (a,b) and (c,d) are equivalent if a+d = b+c, and we write this as (a,b) ~ (c,d). The intuition is that (a,b) represents a-b, so (a,b) and (c,d) are equivalent if a-b = c-d. We can then define an integer to be an equivalence class of ordered pairs of natural numbers; so we define the integer a-b as the set [a,b] := {(c,d) : (a,b) ~ (c,d)}. Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers: [a,b]+[c,d] := [a+c,b+d]; [a,b]·[c,d] := [ac+bd,ad+bc]. The negation (or additive inverse) of an integer is obtained by reversing the order of the pair: -[a,b] := [b,a]. Hence subtraction can be defined as the addition of the additive inverse: [a,b]-[c,d] := [a+d,b+c]. The natural number a may be identified with the integer [a,0].

Rational numbers are defined using ordered pairs (a,b) of integers where b ≠ 0. We say two ordered pairs of integers (a,b) and (c,d) are equivalent if ad = bc, and we write this as (a,b) ~ (c,d). The intuition is that (a,b) represents a/b, so (a,b) and (c,d) are equivalent if a/b = c/d. We can then define the rational number a/b as the set [[a,b]] := {(c,d) : (a,b) ~ (c,d)}. Addition and multiplication of rational numbers can be defined in terms of the equivalent operations on the integers: [[a,b]]+[[c,d]] := [a·d+b·c,b·d]; [[a,b]]·[[c,d]] := [a·c,b·d]. The multiplicative inverse of a rational number is obtained by reversing the order of the pair: 1/[[a,b]] := [[b,a]]. Hence division can be defined as multiplication by this inverse: [[a,b]]/[[c,d]] := [[a·d,b·c]]. The integer a may be identified with the rational number [[a,1]].

In a previous blog post I sketched out a version of natural number arithmetic in which there is a largest natural number m with the following properties: m+a = a+m = m for any natural number a; m·0 = 0·m = 0; and m·a = a·m = m for any natural number a ≠ 0. We can identify m with the integer [m,0] and -m with the integer [0,m]. The integer -m is the smallest number. For any natural number a we have m-a = [m,a] and a-m = [a,m], and therefore m-m = [m,m] = 0. We can identify m with the rational number [[m,1]] and 1/m with the rational number [[1,m]]. The rational number 1/m is the smallest positive number. For any integer a we have m/a = [[m,a]] and a/m = [[a,m]], and therefore m/m = [[m,m]] = 1. We also have m·(1/m) = (1/m)·m = 1 and (1/m)·(1/m) = 1/m.

Other well-known concepts in mathematics can also be constructed from sets. For example, a binary relation between two sets A and B can be defined as a set of ordered pairs (a,b) where a is an element of A and b is an element of b. The set of all such pairs is called the Cartesian product of A and B after the French mathematician René Descartes and is denoted by AxB or A·B. A function from A to B is defined as a binary relation f between A and B that satisfies the following conditions: (1) for all a in A there exists b in B such that (a,b) is in f; and (2) if (a,b) is in f and (a,c) is in f then b = c. The more usual terminology and notation can be derived from this formal definition by writing f(a) = b as shorthand for ‘(a,b) is an element of f.’