[FOM] Hartogs contribution to set theory

[Martin Davis]

As Alasdair Urquhart noted in his recent FOM 
post, the term "Hartogs Theorem" is ambiguous. 
Hartogs is perhaps best known for a result 
concerning the analog of Cauchy's integral 
formula for analytic functions of two complex 
variables. Unfortunately, the recent post by 
Richard Heck with a purported link to Hartogs 
paper on set theory, in fact, pointed instead to 
this work in analysis. I apologize for posting it without checking first.

Here is a link to the correct paper, Hartogs, 
Friedrich: "Úber das Problem der Wohlordnung," 
Mathematische Annalen, 76(1915),436-443:
http://www.springerlink.com/content/g31432862115330n/
The main point of the paper is a proof that the 
principle that cardinal numbers are comparable 
implies (without separate use of the axiom of 
choice) that every set can be well ordered. 
Before this result, it was thought that, although 
AC implies comparability of cardinals, the 
converse was not true. So it was Hartogs who 
first showed that in fact they are fully equivalent.

Martin Davis, moderator