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The well-ordering theorem is a famous result in set theory stating that every set may be well-ordered.
Fundamental for G. Cantor's approach to ordinal arithmetic it was an open problem until E. Zermelo gave a proof in 1904 using the axiom of choice (to which it is in fact equivalent if one admits the principle of excluded middle).
Hence, classically, the well-ordering theorem is one of the many equivalent formulations of the axiom of choice like also e.g. Zorn's lemma.
Given any set $S$, there exists a well-order $\prec$ on $S$.
The first 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 whence the validity of the well-ordering principle became important for his theory of ordinal numbers. In his 1883 paper he calls it a ‘fundamental and weighty law of thought that is remarkable for his generality’ and promised to come back to it later (Cantor 1932, p.169). In the following, he announced proofs but they failed to materialize so that he was forced to take it as an assumption.^{1}
Consequently, the well-ordering principle ended as ‘a very strange claim’ second on Hilbert's millenium list of open problems in mathematics in 1900. Then in 1904, the Hungarian mathematician J. König announced a proof that the continuum could not be well-ordered but had to retract the proof.
Soon afterwards in 1904, Ernst Zermelo finally gave a proof using the axiom of choice following a suggestion by E. Schmidt. The proof, albeit correct, was met with heavy criticism by prominent mathematicians so that Zermelo published a new proof and a defense of the contested axiom of choice in 1908.
The attempt to make explicit the set-theoretic assumptions in the proof led him to publish his axioms for set theory in the same year which became later a part of Zermelo-Fraenkel set theory. Hence the 1904ff controversy proved to become a decisive watershed for the development of modern mathematics: triggering the advent of set-theoretic foundation of mathematics and putting the problem of non-constructive methods of proof on the agenda.
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! (^_^)
The original proof is in
The second proof together with an eloquent defense of the axiom of choice can be found in
Cantor’s texts are collected together with comments by Zermelo in
English versions of Zermelo’s papers are in
On the relation between AC and the well-ordering principle in general toposes see
Peter Freyd, Choice and Well-ordering , APAL 35 (1987) pp.149-166.
M.-M. Mawanda, Well-ordering and Choice in Toposes , JPAA 50 (1988) pp.171-184.
Peter Johnstone, Sketches of an Elephant vol. II, Oxford UP 2002. (D4.5, pp.987-998)
J. Todd Wilson, “An Intuitionistic version of Zermelo’s proof that every choice set can be well-ordered”, J. Symbolic Logic, 66:3 (2001), 1121–1126; (JSTOR: paywalled), formalization in the Lean Theorem Prover.
This contrasts with Cantor’s attitude towards the axiom of choice which he used implicitly but never thematised explicitly. In fact, the full explicit awareness of the use of the axiom choice in mathematics had to await the controversy over Zermelo’s well-ordering theorem in 1904 (with some anticipation by G. Peano and B. Levi earlier). ↩