nLab well-ordering theorem

Idea

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).

Hence the well-ordering theorem is one of the many equivalent formulations of the axiom of choice like also e.g. Zorn's lemma.

Statement and proof

Given any set $S$, there exists a well-order $\prec$ on $S$.

The first proof was given in (Zermelo 1904). Within the nLab, the article Zorn's lemma gives a standard informal proof that can be formalized in ZFC under classical logic, as well as the easy argument that conversely, the well-ordering principle (in its classical “least element” form; see below) implies the axiom of choice and Zorn’s lemma.

Remark

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?

Consequences

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.

In constructive mathematics

In constructive mathematics, the well-ordering principle is also equivalent to the axiom of choice. This was proved by Andrew Swan (2021) and formalized in Agda by Tom de Jong (2021) and in Coq by Dominik Kirst (2021).

Zorn's lemma is, on the other hand, constructively weaker than the axiom of choice, as it doesn’t even imply excluded middle. But together with excluded middle it implies choice. However, Zorn’s lemma is not particularly useful without excluded middle.

Related entries

• Zorn's lemma

• axiom of choice

• Hartogs number

History

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 millennium 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.

References

The original proof is in

• Ernst Zermelo, Beweis, daß jede Menge wohlgeordnet werden kann, Mathematische Annalen 59 (1904) pp.514-516. (gdz)

The second proof together with an eloquent defense of the axiom of choice can be found in

• Ernst Zermelo, Neuer Beweis für die Möglichkeit einer Wohlordnung , Mathematische Annalen 65 (1908) pp.107-128. (gdz)

Cantor’s texts are collected together with comments by Zermelo in

• Ernst Zermelo (ed.), Georg Cantor - Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts , Springer Berlin 1932. (gdz)

English versions of Zermelo’s papers are in

• J. van Heijenoort (ed.), From Frege to Gödel - A Source Book in Mathematical Logic 1879-1931 , Harvard UP 1967.

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.

The proof that Zorn’s Lemma doesn’t imply excluded middle (and hence doesn’t imply choice without assuming excluded middle):

• John Lane Bell, Zorn’s lemma and complete Boolean algebras in intuitionistic type theories, The Journal of Symbolic Logic, 62(4):1265–1279, 1997 (doi:10.2307/2275642)

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1. 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). ↩