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Where can I find the proof of the claim that Zorn's lemma is equivalent to the Axiom of choice?
Landau said:@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?
Landau said: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).SW VandeCarr said: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.)
Landau said:Yes, but I guess I'm missing your point (or maybe your point of your point).
Landau said: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.
Zorn's lemma is a mathematical principle that states that if a partially ordered set satisfies a certain condition, then it must contain a maximal element. This lemma is closely related to the Axiom of Choice, as it can be used to prove the existence of a choice function for any collection of non-empty sets.
Zorn's lemma provides a way to prove the Axiom of Choice by showing that any collection of non-empty sets has a choice function. This is done by constructing a partially ordered set where each element corresponds to a choice function and using Zorn's lemma to show that there must be a maximal element, which is the desired choice function.
The Axiom of Choice is controversial because it allows for the creation of sets that cannot be explicitly constructed or defined. This can lead to counterintuitive results and paradoxes, and some mathematicians argue that it goes against the foundational principles of mathematics.
Yes, Zorn's lemma can be used to prove other theorems in mathematics, particularly in the fields of topology, set theory, and functional analysis. It has also been used to prove the existence of solutions to various optimization problems.
Yes, there are several alternative principles that have been proposed as substitutes for the Axiom of Choice, such as the Axiom of Determinacy and the Axiom of Global Choice. These principles aim to avoid some of the counterintuitive consequences of the Axiom of Choice while still allowing for the creation of choice functions for collections of sets.