* Order Structures

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* Order Structures

In this chapter we develop one of the fundamental structures in modern mathematics: the order structures. We will consider three types of orders: partial orders, total orders, and well--orders. More attention is given to well--orders. We will prove the transfinite recursion theorem for well--orders, and then use it to construct the "ordinal numbers", with which we can complete our definition of cardinal numbers of sets.

In this chapter, three new axioms are added into our axiom list, namely replacement axioms, choice axiom and regularity axiom. We have made use of a specific form of the axiom of choice in previous chapters, but there is much more. We will see there are a number of equivalent forms of the axiom of choice.

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