In this chapter we investigate a particular class of well-founded relations, namely those which are linear orderings. We begin with a few definitions.

Let < be a binary relation on a set a. We say

1. < is irreflexive if (∀x∈a)¬(x < x);

2. < is antisymmetric if (∀x, y∈a)((x < y) ⇒ ¬(y < x));

3. < is transitive if (∀x, y, z∈a)(((x < y) ∧ (y < z)) ⇒ (x < z));

4. < is trichotomous if (∀x, y ∈a)((x < y) ∨ (y < x) ∨ (x = y));

and < is a linear (or total) order on a if it satisfies all the above conditions. Actually, the second condition is redundant, since it is implied by the first and third. Moreover, a well-founded relation is always irreflexive (since if x < x then {x} has no <-minimal member), and a well-founded trichotomous relation is transitive (since if we have x < y and y < z but not x < z, then {x, y, z} has no minimal member). Thus a well-founded relation is a linear order iff it is trichotomous; we call such a relation a well-ordering of the set a. Equivalently, a well-ordering of a is a linear ordering < of a such that every nonempty subset of a has a (necessarily unique) <-least member. (We say that x is the least member of a subset b, rather than simply minimal, if (∀y∈b)((x < y) ∨ (x = y)).)