Hello, infinite-set reader. Set theory is back. Today we will learn about Well Ordering Theorem.
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.
Well Ordering Theorem
(x) Theorem :
Every poset has largest simply ordered subset.
Given poset . Consider poset with is the set of all simply ordered subset of poset . By Theorem in the previous post then every simply ordered subset of poset have smallest upper bound. Since is poset that the every simply ordered subset has smallest upper bound then by Zorn’s Lemma, has at least one maximum element. Suppose that the maximum element of is then is also largest simply ordered of poset .
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.
Take any set of . Consider set is the set of all subsets of that can be well ordered. Then is partially ordered by . By Theorem (x) above then has largest simply ordered subset . Claim that is well ordered and .
For proof is well ordered analogue on proof of theorem in this link.
Next will be proved . It is clear that because for all it is true that . Next, for proof , suppose not to apply so then it exists such that . Consider then can be well ordered with is the largest/last element of . As a result is simply ordered subset of which contains proper subset . Contradictions with is largest simply ordered subset of . Proven can be well ordered.
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