3Posets and Zorn's lemma

II Logic and Set Theory



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.

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