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.

  • $\begingroup$ 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 $\endgroup$
    – nwr
    Jan 21, 2017 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, 2017 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, 2017 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! $\endgroup$
    – nwr
    Jan 23, 2017 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, 2017 at 11:38

1 Answer 1


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.


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