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.
For any poset with the every non-empty well ordered subset has the smallest upper bound on , then has at least one maximum element.
Given is poset with the every non-empty well ordered subset has the smallest upper bound on . Take any function with for all . Next, take any . By Theorema (post sebelumnya) then can be consider well ordered set such that bounded from above and . It will be shown that is one of the maximal elements of . Take any . If or then it is always true that “if then ”. Also if and then it is always true that “if then ”. In other words, is one of the maximal elements of .
Next we will discuss one of the theorems. But before discussing the theorem, will be given a new definition of the symbol which also produces the following theorem.
Given is the set of all well ordered subsets of poset . Then can be partial order by by definition, for any :
It will be shown that is a partial order.
Clearly that then for all . It’s proved reflexive.
Next, it will be proved that if and then . Suppose then from there is such that and from then exist such that . So, and . Obtained and which both statements contradict each other. It’s proved antisymmetric.
Next for transitive. If and then it can be easily proven that . I deliberately leave evidence for this point for you to try.
Because of is a partial order, then for any is the set of all well ordered subsets of poset , Q can be partial order by .
See you on my next post. The next post will use the definition above. So do not forget. Thanks for reading. May be useful.