Ultrafilters - when did it start?

Question

I am writing a paper on some of the applications of ultrafilters, especially the ones on $\mathbb{N}$. I thought that it would be interesting to include some information about how the concept got introduced into mathematics. I would suspect that the earliest applications would be in topology or in Boolean algebras, but I can't seem to find any historical reference. Could someone be so kind as to offer a reference to when and how ultrafilters were first used, or some directions of where I should start digging?

(Neither The Theory of Ultrafilters" by Comfort & Negrepontis, nor "Algebra in the Stone-Čech Compactification" by Hindman & Strauss seem to contain much reference to history of the concept).

Martin Sleziak · Answer 1 · 2013-07-10T20:16:36.743

The notion of filter was defined for the first time probably by Henri Cartan [Cart1] (see also [B, §6.1]). However, some authors mention that the notion of filterbase was used by L. Vietoris [V] under the name "Kranz" prior to H. Cartan (see [P, p.46], [R1] or [R2]). R. M.Dudley [D, p.76] mentions that C. Caratheodory used filterbases in [Cara1] and M. H. Stone was in fact dealing with filters in [S].

[B] N. Bourbaki. Elements of Mathematics. General Topology. Chapters I-IV. Springer-Verlag, Berlin, 1989.
[Cara1] C. Carathéodory. Über die Begrenzung einfach zusammenhängender Gebiete. Math. Ann., 73:323-370, 1913.
[Cart1] H. Cartan. Théorie des filtres. C. R. Acad. Sci. Paris, 205:595-598, 1937.
[Cart2] H. Cartan. Filtres et ultrafiltres. C. R. Acad. Sci. Paris, 205:777-779, 1937.
[D] R. M. Dudley. Real Analysis and Probabilty. Cambridge University Press, Cambridge, 2002.
[P] G. Preuss. Foundations of Topology. Kluwer, Dordrecht, 2002.
[R1] H. Reitberger. The contributions of L. Vietoris and H. Tietze to the foundations of general topology. In C. E. Aull and R. Lowen, editors, Handbook of the history of general topology, Volume 1, pages 31-40. Kluwer, Dordrecht, 1997.
[R2] H. Reitberger. Leopold Vietoris (1891-2002). Notices of the American Mathematical Society, 49(10):1231-1236, 2002.
[S] Marshall Harvey Stone. The theory of representations for Boolean algebras. Trans. Amer. Math. Soc., 40:37-111, 1936.
[V] L. Vietoris. Stetige Mengen. Monatshefte f. Math., 31:545-555, 1921.

@Martin Thanks a lot for editing my post and including the links! (I should have not been so lazy.) — Martin Sleziak, Jun 22 '13 at 14:55

score 11 · Accepted Answer · edited Jun 20 '13 at 12:42

11

The existence of ultrafilters was first proved by Tarski in

Tarski, A.: Une contribution à la théorie de la mesure, Fundamenta Mathematicae 15 (1930), 42-50.

This is mentioned in a recent article here.

edited Jun 20 '13 at 12:42

Did

284,245

answered Jun 20 '13 at 12:28

Mikhail Katz

47,573

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Tarski's paper can be found here. – Asaf Karagila Jun 20 '13 at 12:49
Thank you! It is exactly the kind of answer I was hoping for. – Jakub Konieczny Jun 20 '13 at 19:23

Andreas Blass · Answer 3 · 2013-06-17T00:13:35.110

9

I believe the concept of ultrafilter, but not the name, first arose in the work of Stone on the topological representation of Boolean algebras. After all, the points of the Stone space of a Boolean algebra are the ultrafilters in that algebra. The name "ultrafilter" was probably introduced by Henri Cartan, perhaps via Bourbaki.

edited Jun 17 '13 at 00:13

answered Jun 14 '13 at 20:17

Andreas Blass

75,557

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About Cartan, see Théorie des filtres. CR Acad. Paris, 205, (1937) 595–598 and Filtres et ultrafiltres. CR Acad. Paris, 205, (1937) 777–779. – Seirios Jun 14 '13 at 21:27
Note that Bourbaki meetings did not start until the mid-1930s. Tarski's paper dates from 1930. Cartan reportedly used the term "boum" for it. – Mikhail Katz Jun 20 '13 at 12:29

Ultrafilters - when did it start?

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