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Basic of Zorn’s Lemma – Initial Segment

Hello, infinite-set reader. Set theory is back.

In this post will be discussed about a theorem that will become the basic of Zorn’s Lemma. To remind you of initial segment, you should visit this post first. The definition of the initial segment will be used later.

Theorem :

For any poset \left(P,\leq\right) with the every non-empty well ordered subset has the smallest upper bound on P, then P has at least one maximum element.

Proof :

Given \left(P,\leq\right) is poset with the every non-empty well ordered subset has the smallest upper bound on P. Take any function f:P\rightarrow P with x\leq f\left(x\right) for all x\in P. Next, take any p\in P. By Theorema (post sebelumnya) then can be consider well ordered set W\left(p\right) such that W\left(p\right) bounded from above and m=sup\, W\left(p\right)\in W\left(p\right). It will be shown that m is one of the maximal elements of P. Take any x\in P. If x\in W\left(p\right) or x\leq p then it is always true that “if m\leq x then x=m”. Also if x\notin W\left(p\right) and x\nleq p then it is always true that “if m\leq x then x=m”. In other words, m is one of the maximal elements of P.

Next we will discuss one of the theorems. But before discussing the theorem, will be given a new definition of the symbol \ll which also produces the following theorem.

Theorem :

Given Q is the set of all well ordered subsets of poset \left(P,\leq\right). Then Q can be partial order by \ll by definition, for any u,v\in Q :

    \[ \left(u\ll v\right)\equiv\left(u\mbox{ is the initial segment of }v\mbox{ or }u=v\right)\]

Proof : (more…)