This is another rant on Zorn’s lemma. And hopefully the final one. It is something that has puzzled me for more than a year now!
In a partially ordered set , let every chain have an upper bound. Why is it not obvious that has a maximal element?
The whole concept depends on the following construction: can we construct an ascending chain such that for every element in the chain, if there is a greater element(s) in , then we add one of those greater elements to the chain after ? For example, take to be , and construct the chain . We know that , all these elements are greater than . Hence, we can choose any of those greater elements and add it to the chain, we can form a bigger chain.
If we continue adding such greater elements to the chain until we cannot add anymore, then the greatest element is clearly the maximal element of the whole of , and not just the chain. Because is was not the maximal element, then we could add another element from in front of in the chain, contradicting the fact that we cannot add any other element to the chain.
Rephrasing the earlier question: if in a partially ordered set every chain has an upper bound, can we prove the existence of a chain in which for every element in the chain if a greater element exists, one of those elements has been added to the chain? Other similar questions would be: can we prove the existence of a chain in which is isomorphic to , or can we prove the existence of a chain which contains all numbers between and ? The point to be taken away from this is that proving the existence of such chains is not trivial or “obvious”.
Where does Zorn’s lemma come in? Zorn’s lemma allows us to make an infinite number of arbitrary decisions. For example, if we have infinite pairs of shoes, Zorn’s lemma would allow us to arbitrarily choose one shoe from each pair in that infinite set, without detailing which shoe we picked from each pair, or how we went about choosing the shoe. Making this more clear, picking the left (or right) shoe from each pair does not require Zorn’s lemma, as the choice made for each pair is explicitly clear. However, selecting any *random* show from each of the infinite pairs requires the use of Zorn’s lemma, as we’re making an infinite number of decisions without really giving details.
How is this relevant to proving the existence of the aforementioned chain? As we can make an infinite (possibly uncountable) number of arbitrary choices without detailing how, we can, for each element in the chain with a greater element in , add one of those elements (this is the decision making part) to the chain, and continue doing so infinitely.
Note that for each element in the chain, if the set of greater elements in had a lowest greater element or something similar, we wouldn’t need Zorn’s lemma. We could just state “we choose the lowest greater element”. However, such a distinct element in the set of elements greater than need not exist in every partially ordered set . Hence we need Zorn’s lemma.
Note that Zorn’s lemma does not imply that this chain has an upper bound in ! However, if it does, as assumed in the statement of the lemma, then that will undoubtedly be a maximal element of .