Well ordering principle

This week we talked about the well ordering principle present in geometric and arithmetic sequences.

The statement is: 

For every set X, there exists a well-ordering with domain X.

How do we prove it then?

Take the set A of all well-orderings of subsets of X: an element of A is an ordered pair (a,b) where a is a subset of X and b is a well-ordering of a. A can be partially ordered by continuation. That means, define E ≤ F if Eis an initial segment of F and the ordering of the members in E is the same as their ordering in F. If E is a chain in A, then the union of the sets in E can be ordered in a way that makes it a continuation of any set in E; this ordering is a well-ordering, and therefore, an upper bound of E in A. We may therefore apply Zorn's Lemma to conclude that A has a maximal element, say (M,R). The set M must be equal to X, for if X has an element x not in M, then the set M∪{x} has a well-ordering that restricts to R on M, and for which x is larger than all elements of M. This well ordered set is a continuation of (M,R), contradicting its maximality, therefore M = X. Now R is a well-ordering ofX.

In 9.1 about sequences and summation notation, we can see that every set of numbers can be organized because there is a relationship existing in between. 

 Example:

Find the relationship between 1,3,5,7... and 1,2,3,4

Fathering putting the two sequences side by side, we can see the relationship 

An = 2n-1