Main Question or Discussion Point
Where can I find the proof of the claim that Zorn's lemma is equivalent to the Axiom of choice?
I may be mistaken, but the well ordering theorem would seem to be implicit in any proof that requires the identification of a minimal or maximal element.@SW VandeCarr: The well-ordening theorem, ZL, AC, and e.g. the "Principle of Cardinal Comparability" (for any sets A and B we have |A|<=|B| or |B|<=|A|), are in fact all equivalent.
I don't see how Fredrik would be implicitly using the well-ordening theorem. Could you please elaborate?
OK. I would argue that the comparability of pairs is a consequence of the well-ordering theorem (WOT). However, my real point was that, as you say; AC, ZL and WOT are all set theoretically equivalent. Therefore, logically one being true iimplies that the others are true. If one were false, the others would be false. (I'm using "true" in the sense: If P and Q, then P->Q.)Yes, you are mistaking. In a typical proof using Zorn's Lemma, you have to show that every chain has an upper bound. Often the partial order is just set inclusion with function restriction: pairs (f,A) where f is a function with domain A, (f,A)<= (g,B) iff A is contained in B and the restriction of g to A equals f. In this case, a chain is a set of pairs (f_i,A_i) which are all comparable; it has as upper bound (h,X), where X is the union of all sets in the chain, and h is the (unique) function whose restricition to A_i equals f_i. This is just basic stuff about functions and sets, and has nothing to do with the well-ordening theorem.
Yes, but I guess I'm missing your point (or maybe your point of your point).However, my real point was that, as you say; AC, ZL and WOT are all set theoretically equivalent. Therefore, logically one being true iimplies that the others are true. If one were false, the others would be false. (I'm using "true" in the sense: If P and Q, then P->Q.)
I was simply putting some context around Fredrik's question,which was narrowly posed. It can be argued that WOT is more fundamental than AC or ZL. The important fact, IMHO is not that you can prove AC from ZL but that there is a larger set of propositions that are all based on the the assumption of a R(a,b) such that a set is well ordered.Yes, but I guess I'm missing your point (or maybe your point of your point).
Saying Frederik's question was narrowly posed was not a criticism. Also, I admit "more fundamental" is not a precise statement. I mean that the well ordering of the rational numbers can be proven within ZF, without C. To prove the well ordering of the reals you need AC, but AC can be proven from WOT. WOT could just as well be an axiom of ZF. However, I do have a question about this and will post a new thread.I'm still not following. We agree that AC, ZL, and the WOT are all logically equivalent. Why, then, would the WOT be more fundamental than the others, or why should we regard AC and ZL as consequences of the WOT? Surely there's no logical reason, or are you talking about philosophy?
Besides, I think Fredrik's question is as clear as it can get. Whatever the context, it can't be misinterpreted.