1
$\begingroup$

Who was the first mathematician to define and investigate measurable cardinals? And how long did it take for the fundamental theorem of measurable cardinals to be hypothesized and proven?

I would be very grateful for some background information on these questions.

$\endgroup$
  • $\begingroup$ According to this paper, it was Stansilaw Ulam who first defined measurable cardinals and investigated their properties in his 1930 paper Zur Maßtheorie in der allgemeinen Mengenlehre., Source : mathreview.uwaterloo.ca/archive/volii/2/measure-cardinality.pdf $\endgroup$ – Nick R Jan 21 '17 at 17:07
  • 1
    $\begingroup$ What is the "fundamental theorem of measurable cardinals"? Is it Ulam's proof of their weak inaccessibility, Solovay's equiconsistency theorem, Keisler-Scott's critical point characterization, incompatibility with the axiom of constructibility, or something else? $\endgroup$ – Conifold Jan 21 '17 at 21:31
  • $\begingroup$ I mean Keisler-Scott's critical point characterization, i. e. a cardinal is measurable iff it is the critical point of a non-trivial elementary embedding of the universe V into a transitive class M. $\endgroup$ – C. Maier Jan 21 '17 at 23:18
  • $\begingroup$ This paper on the Stanford site appears to contain exactly what you are looking for in a readable summary form :www-logic.stanford.edu/seminar/1516/Mieczyslaw_1103.pdf starting with Banach and Ulam's definition and leading to Keisler's theorem via Scott's results together with those of Łoś and Mostkowski. It even has a cool picture of Bullwinkle J. Moose! I would have summarised it for you here, but that might give the impression that I fully understood it! $\endgroup$ – Nick R Jan 23 '17 at 5:21
  • $\begingroup$ @NickR Thanks a lot! I would love to have some dates, and especially some white sources, but that's already great! How did you found that? $\endgroup$ – C. Maier Jan 23 '17 at 11:38
2
$\begingroup$

You can see : Dov Gabbay & Akihiro Kanamori & John Woods (editors), Handbook of the History of Logic. Volume 6: Sets and Extensions in the Twentieth Century (2012), especially :

  • Ch.1 : Set Theory from Cantor to Cohen, By A.Kanamori,

  • Ch.2 : History of the Continuum in the 20th Century, by J.Steprans,

  • Ch.4 : Large Cardinals with Fprcing, by A.Kanamori.

Also : Akihiro Kanamori, The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed., 2009) : Introduction and Chapter 1. Beginnings.

Maybe useful also : Matthew Foreman & Akihiro Kanamori (editors), Handbook of Set Theory (2010) : Introduction by A.Kanamori.


For historical context, you can see also : Gregory Moore, Zermelo's Axiom of Choice : Its Origin Development and Influence (1982): Ch.4 The Warsaw School.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.