1. Initial segments, well ordering and the axiom of choice. 1.1. Initial segments. We suppose throughout this subsection that < linearly 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 Collections: Mathematics