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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

  1. 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?

Martin Sleziak
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Pecan Lim
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    How do you know that the chain has no upper bound? I think you want to say "well if it did have an upper bound, just keep going", but there's a question of what "keep going" means precisely. Typically what that would mean is you have some fixed well-ordered set you are doing recursion over, so then what is that well-ordered set? – Eric Wofsey Nov 26 '23 at 00:04
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    That said, there are ways of formulating the proof that avoid going through the proper class of all ordinals. In fact it suffices to go up to the Hartogs number, the smallest ordinal that can't inject into your set. (And you don't even need to have the Hartogs number as a von Neumann ordinal--you can just construct a well-ordering of that length explicitly. See this answer for instance.) – Eric Wofsey Nov 26 '23 at 00:09
  • Hmmm... i'm starting to see it now. The ordinals is the only "set" where every chain has an upper bound, and yet no maximal element. But we define that to be not a set, bravo! – Pecan Lim Nov 27 '23 at 05:52

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