 
Summary: 1. Initial segments, well ordering and the axiom of choice.
1.1. Initial segments. We suppose throughout this subsection that < linearly
orders the set X.
Definition 1.1. A subset I of X is an initial segment if
x X, y I and x < y x I.
Trivially, and X are initial segments.
Proposition 1.1. The union of a family of initial segments is an initial segment.
The intersection of a nonempty family of initial segments is an initial segment.
Exercise 1.1. Prove this Proposition. If you understand the union and intersection
of families of sets its simple.
Proposition 1.2. Suppose I and J are initial segments. Then either I J or
J I.
Proof. It will suffice to show that
(1) J I = I J.
So suppose y J I and, contrary to (1), there is x I J. Since x = y we
have either (i) x < y or y < x by trichotomy. If x < y we have x J since J is an
initial segment and if y < x we have y I since I is an initial segment, neither of
which is possible since x J and y I. Thus (1) holds.
Remark 1.1. let I be the family of initial segments. For I, J I declare
I J
