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Well Ordering Theorem

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

This post is the last post about Zorn’s Lemma. The next post will be different from the previous post. But in the end, all the discussion of set theory will be interrelated. My imagination, our discussion will lead to the discussion of cardinality. Pray I write diligently. Happy reading.

(x) Theorem :

Every poset has largest simply ordered subset.

Proof :

GivenĀ  poset \left(P,\leq\right). Consider poset \left(Q,\subseteq\right) with Q is the set of all simply ordered subset of poset \left(P,\leq\right). By Theorem in the previous post then every simply ordered subset of poset \left(Q,\subseteq\right) have smallest upper bound. Since \left(Q,\subseteq\right) is poset that the every simply ordered subset has smallest upper bound then by Zorn’s Lemma, Q has at least one maximum element. Suppose that the maximum element of Q is r then r is also largest simply ordered of poset P.

As already mentioned, the following theorem explains that for any set, the set can be formed as a well ordered set. This theorem is often called the Well Ordering Theorem.

Theorem : (Well Ordering Theorem)

Every set can be well ordered.

Proof (more…)