[FOM] Hartogs contribution to set theory
Martin Davis
eipye at pacbell.net
Sun Nov 7 20:08:22 EST 2010
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
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