[FOM] Cardinals and Choice

[Ali Enayat]

Richard Heck (Dec 22) has written:

>All the helpful books are at the office, so I'll ask here: Does the
>principle that the cardinals are well-ordered imply the axiom of choice?

Indeed, already Trichotomy implies AC; this key result is due to
Hartogs (1915), who showed that if for all sets A and B, there is
either an injection from A to B or vice versa, then the axiom of
choice holds.

Most modern accounts of this result take advantage of Replacement in
the interest of concision of exposition, but the result is well-known
to be already provable in Zermelo set theory (without recourse to von
Neumann ordinals, which were introduced in 1920's).

The following previous FOM postings of Davis and Urquhart are also
useful; the first one (Davis) has a link to Hartogs' original paper
(in German); the second one (Urquhart) re-iterates the important fact
that the result of Hartogs is provable in Zermelo set theory, and
points out a relevant historical paper by Greg Moore on the subject.

http://cs.nyu.edu/pipermail/fom/2010-November/015119.html

http://cs.nyu.edu/pipermail/fom/2010-November/015120.html

Regards,

Ali Enayat