Why Read This Book? - Index of

118 Chapter 3 Sets and Their Properties
EXERCISE 3.11.24 The least element of a well-ordered set is unique.
EXERCISE 3.11.25 Show that a well ordering is a total ordering by showing
that property W4 implies T4.
Though a well ordering is a total ordering, a total ordering is not necessarily a
well ordering. For example, ≤ is a total ordering of the integers that is not a well
ordering, for the entire set of integers does not contain a least element. Since W4
implies T4, but not vice versa, property W4 is therefore stronger than T4.
Just because a given a set with an order relation fails to be a well ordering,
it does not mean that the set cannot be well ordered by some other relation. In
fact, one of the most notable results in the modern theory of sets is that any set
can be well ordered. Ernst Zermelo demonstrated this in 1904, using an axiom of
set theory that we have said nothing about so far in this text. The axiom of choice
is a somewhat mysterious axiom of set theory that is simple to state but seldom
addressed at this level of the mathematical game. It says, “Given any family F of
mutually disjoint non-empty sets, there is a set S that contains a single element
from each set in F.” Thus S can be thought of as the result of having chosen a
distinct representative element from each set in F. In the axiom of choice, there is
enough strength to demonstrate that for any set, there exists a relation on A × A
that is a well ordering of A. One way to see how the integers can be well ordered
is to list its elements as 〈0, −1, 1, −2, 2, −3, 3, −4, 4,...〉 and define xRy if x does
not come after y in this listing. When the integers are ordered in this way, every
non-empty subset has a least element.