I'm not understanding why its necessary to invoke the knowledge about ordinals in order to prove Zorn's lemma.
The Hypothesis in Zorn's lemma is
- Every chain in the set Z has an upper bound in Z
Then Z has a maximal element.
The sketch of the proof by contradiction goes as: Assume Z has no maximal element. Take a chain in Z, it has an upper bound. Add that upper bound to the chain to get a new chain. Since Z has no maximal element, you can find a new upper bound, add that to the chain too. Keep going forever, eventually your chain is as long as the ordinals, which is not a set (its too big).
I find it unnecessary to invoke the fact about ordinals being too large to be a set. Even if we did not know that, the fact that you can keep adding upper bound elements to the chain means you have found a chain with no upper bound, thereby contradicting hypothesis 1). So why can't we stop here in the proof, since we already have a contradiction? Why do we need to go further and say something about the size of the chain being too long?