I know that the Zorn's lemma (If every totally ordered subset of a partially ordered set S has an upper bound, then S contains a maximal element) can be used to prove the existence of a maximal element.
However, I need to prove that a certain partially ordered set S, which contains infinite (countable) elements, has no maximal element.
Now I suppose that the reverse of Zorn's lemma is not necessarily true (that is: If a partially ordered set S contains a maximal element, then every totally ordered subset of S has an upper bound). Mine is just a supposition because I cannot figure out a counterexample, but my gut instinct – and the fact that I always found Zorn's lemma statement written as an "if – then" and never as an "if and only if" – is telling me that the reverse is not true.
Now I now that the Zorn's lemma is equivalent to: "If a partially ordered set S doesn't contain a maximal element, then there exists a totally ordered subset of S which has no upper bound." but this is not helping me in my proof.
So the question is: what valid arguments (if any) can be used to demonstrate that a certain poset has no maximal element? There is another well-known "matematician tool" – that can lead to the conclusion I'm searching for?
Best Answer
It's impossible to say in generality what form the argument would take, but I think often one would be able to use proof by contradiction (suppose your poset has a maximal element, then show why a contradiction follows.) As a simplified example "Suppose $\mathbb N$ has a maximal element $n$. But then there's $n+1$ which is greater than $n$, oops!"
Zorn's lemma doesn't really help you at all with respect to this task. From the statement you have something that concludes there is a maximal element, and from the contrapositive you conclude there must be an infinite ascending chain. But what you want is something that concludes "there is not a maximal element." So, an argument other than what makes Zorn's Lemma work is called for.
I think if you need more advice you'll have to share what poset you're thinking of.