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The well-ordering theorem states that every set may be well-ordered. It is a theorem in the sense that it can be proved assuming the axiom of choice. On the other hand, the axiom of choice can instead be proved using the well-ordering theorem and the principle of excluded middle.
Given any set $S$, there exists a well-order $\prec$ on $S$.
The proof was given in (Zermelo 1904).
If every set can be well-ordered, then the natural map from ordinal numbers to cardinal numbers is a surjection. Since these form proper classes, the ordinary axiom of choice will not split this surjection automatically; however, we can easily split it by assigning each cardinal number to the smallest ordinal number with that cardinality. (This does require excluded middle, however.) In this way, a cardinal number may be defined to be an ordinal number that is initial: such that no smaller ordinal number has the same cardinality.
As in the argument above, the axiom of choice follows; given any surjection $f: A \to B$, place a well-ordering on $A$ and then split $f$ by mapping an element $y$ of $B$ to the smallest element $x$ of $A$ such that $y = f(x)$. Again, this uses excluded middle to show that such a smallest element exists, so the well-ordering principle does not (seem to) imply the axiom of choice constructively.
To get the large (or “global”) axiom of choice (that any surjection between proper classes splits), we need a large well-ordering theorem: that every proper class can be well-ordered. The large principles do not follow from the small ones.
Georg Cantor first developed set theory in the context of studying well-ordered sets of real numbers and considered the well-ordering theorem a ‘law of thought’ (‘Denkgesetz’) without need of proof.
Ernst Zermelo gave a proof in 1904, but many mathematicians were concerned about the proof's validity. Analysis of Zermelo's proof isolated the axiom of choice and led to that axiom's first explicit formulation (also by Zermelo).
That the well-ordering theorem is more of a theorem in need of a proof, while the axiom of choice is more of an axiom to be assumed without proof is, of course, a matter of opinion, but it's reflected in Jerry Bona's famous quotation:
The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?
Ironically, in constructive mathematics, the well-ordering principle is (seemingly) actually weaker than the full axiom of choice, as it does not imply excluded middle by itself. It does, however, imply the full axiom of choice (and hence excluded middle) if by ‘well-order’ we mean a classical well-order, in the sense that every inhabited subset has a least element, rather than the constructively sensible notion of well-order that merely permits inductive proofs. Zorn's lemma is likewise constructively weaker than the axiom of choice, although it is not particularly useful without excluded middle.
At least, as far as I can tell it doesn't. I've never actually seen a metamathematical result proving this, however.
Mike: If “well-order” is interpreted in the classical sense “every nonempty set has a least element,” then I believe it does imply full choice. I don’t know what happens if “well-order” is interpreted in the constructively reasonable sense. I know that Zorn’s lemma, even in the classical version, can be true in a non-Boolean topos, although it is not particularly useful without excluded middle.
Toby: That's correct; any well-ordered set, even in the slightly weaker sense that every inhabited subset has a least element, must be a choice set (as you remarked at choice object). One of us should add a note about that to this article.
Mike: It would be nice to have different words to distinguish between “constructively correctly well-ordered” and “classically well-ordered.” The Elephant, at least, uses “well-ordered” for the classical notion which implies choice; do most practicing constructivists use the constructively sensible notion?
Toby: Shockingly many constructivists aren't aware how to fix the definition (which is highly non-predicative anyway). Some use the no-infinite-descending-chains version for the purpose of doing Burali-Forti, but it's not good for anything else. Since the existence of any ‘classically’ well-ordered set with two non-equal elements implies excluded middle already, many constructivists consider the concept useless. So basically, constructivists either have to do it right or not at all.
We have ‘choice set’ for ‘classically well-orderable set’, so maybe we could say ‘choice order’ or something for ‘classical well-order’. (Obviously, those constructivists who say ‘choice set’ for a projective set wouldn't use this term.) There may be more terms in Paul Taylor's paper on various kinds of ordinals in constructive mathematics (as isomorphism classes of well-ordered sets are not the only useful kind), which I've never fully read.
But I think that ‘classically well-ordered’ should be all right. By default, ‘well-ordered’ should mean the constructively correct notion, since this is what constructivists need and classical mathematicians don't care. The only place where ‘classically well-ordered’ needs to be said is here (and possibly at axiom of choice), in the definition section of well-order, and perhaps in statements about well-ordered sets that aren't constructively valid (if adding ‘classically’ will make them constructively valid), but even that is kind of pointless and it's simpler just to say that the result is not constructively valid and useful.
Mike: I’ve started trying to uniformize terminology among these pages, using ‘classical well-order’ where appropriate. I agree that ‘well-order’ should mean the constructively correct notion, but I don’t want to alienate non-constructivists by hiding the definition they are familiar with in comments about constructivism. So I’m trying to isolate the comments about constructivism a bit. I think this query box could probably be deleted now.
Toby: Well, constructivists are used to being isolated. At least we give the good definitions first, so that's a big improvement! (^_^)