The Construction of the Real Numbers

Abstract

Commutative laws: Addition and multiplication are commutative operations. Associative laws: Addition and multiplication are associative operations. Distributive law: x(y + z) = xy + xz as usual. Identity for multiplication: 1 is the identity element for multiplication. Inverse law for multiplication: Every number a has a multiplicative inverse a. (recall that zero is not in our system). Definition: We define x < y as ∃z(x + z = y) then >, ≤, ≥ are defined using < as usual. Trichotomy 1: For any x and y, either x < y, y < x, or x = y. Trichotomy 2: For any x and y, either ¬x < y or ¬y < x.