3 Posets and Zorn’s lemma

In this chapter, we study partial orders. While there are many examples of

partial orders, the most important example is the power set

P

(

X

) for any set

X

,

ordered under inclusion. We will also consider subsets of the power set.

The two main theorems of this chapter are Knaster-Tarski fixed point theorem

and Zorn’s lemma. We will use Zorn’s lemma to prove a lot of useful results in

different fields, including the completeness theorem in propositional calculus.

Finally, we will investigate the relationship between Zorn’s lemma and Axiom of

Choice.