Erhard Scholz, in his article "Felix Hausdorff and the Hausdorff edition" writes the following:

"Hausdorff considered the contemporary attempts to secure axiomatic foundations for set theory as premature. Working on the basis of a 'naive' set theory (expressedly understood as a semiotic tool of thought), he nevertheless achieved an exceptionally high degree of argumentation."

I have some questions regarding the above quote:

  1. Is there anything to Hausdorff's view that the attempts to secure axiomatic foundations for set theory were premature (of course modern set theorists have, by their actions, essentially said 'no', but it still might be interesting to reconsider the question)? I take the term "premature" to mean that axiomatizing a certain area of mathematics 'freezes' mathematical practice in that area for critical analysis, such as producing independence results and that Hausdorff's view (possibly) was that the attempts of his contemporaries to secure axiomatic foundations for set theory were premature in that current mathematical practice regarding set theory was not mature enough to provide adequate axiomatizations (of course Godel's results show that any attempt to axiomatize set theory might arguably be 'premature').

  2. If Hausdorff worked in a 'naive' set theory "expressedly (Scholz's term) understood as a semiotic tool of thought" (whatever that means), how did he resolve the paradoxes in his 'naive' set theory?

  3. Can one adequately resolve the paradoxes of naive set theory (Russell's, Burali-Forti, and Cantor's, etc.) via semiotics and if so, why is no one seeming to work in this 'reformed' naive set theory?

  • 1
    $\begingroup$ I don"t think anyone here will be able to answer your current spate of questions, primarily as semiotics is (in my view) not an area of active mathematical research directly. If you were able to provide technical data on what Hausdorff did do, or something that Hausdorff did write, that kind of question might get a good response. Interesting as your question is, I am not optimistic that this forum will see it answered. Gerhard "Ask Me About System Design" Paseman, 2012.04.14 $\endgroup$ Apr 14 '12 at 10:51
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    $\begingroup$ Reagrding 2 and 3, I am quite sure that you are overinterpreting something here. How to resolve Russell's paradox? Well, just don't form such a weird set for which you anyway have no need in 'standard mathematical practise'. And, also today, most uses of set theory are naive, a language to phrase things (semiotic tool?). I hardly need the axiom of regularity to tell me there is no infinite descending chain relative to 'is an element'. For most sets I ever encounter this is 'obviously' true. (ps. This is not meant as dismissive versus axiomatic set theory, which I find interesting.) $\endgroup$
    – user9072
    Apr 14 '12 at 11:57
  • $\begingroup$ find this question to be extremely interesting. Another possible (in case Gerhard is right) forum that you might try is philosophy.stackexchange.com . If you migrate their let us know. I for one would be interested in following the discussion. $\endgroup$ Apr 14 '12 at 23:14
  • $\begingroup$ Thanks for the comments. I was hoping that there might be someone on Math Overflow who has deep familiarity with Hausdorff's work who might be able to answer the questions. I looked through his book on set theory (the English translation) and found no comments on the paradoxes. Also, his early chapters on sets seem to me to refer to sets of (or at least could be construed to be sets of) urelements. As far as I know (perhaps I am not being enterprising enough) one cannot derive contradictions using Naive Set Theory when one is referring to sets of urelements. $\endgroup$ Apr 15 '12 at 6:44
  • $\begingroup$ I'm not sure why Scholz brought semiotics into the mix if Hausdorff did not specifically consider semiotics (the theory of signs) in any of his work (published or unpublished) on set theory. What I find most interesting is Hausdorff considered the attempts of Zermelo and others to axiomatize set theory in order to rid it of the paradoxes (at least the Big Three-- Russel's, Burali-Forti's, and Cantor's) premature. If Sholz is correct then he must have had something to say about the paradoxes or at least an argument as to why one could 'mature' $\endgroup$ Apr 15 '12 at 7:02

I'll attempt an answer to question 1. Hausdorff was entitled to think that set theory was not yet mature, because his own 1914 book made considerable advances on what had been done previously (notably by Cantor and Zermelo). It is worth reading the glowing review in the 1920 Bulletin of the AMS to see how his book changed the perception of set theory by mathematicians. Just to mention two of his contributions: definition of a topological space, and the paradoxical decomposition of the sphere that paved the way for the Banach-Tarski paradox.

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    $\begingroup$ The original axiomatization of set theory missed foundation and replacement, which are key components of our current intuitive picture of the set theoretic universe. The Zermelo-Fraenkel list misses large cardinals. And so on. Perhaps the idea that the axiomatization was premature was not that off the mark. $\endgroup$ Jan 26 '13 at 2:48
  • $\begingroup$ @John: Nice answer. I looked through Blumberg's review and thoroughly enjoyed it! Another good article along the same line is Peter Koepke's "Felix Hausdorff and the Foundations of Mathematics". In it, regarding the paradoxes, Koepke quotes Hausdorff as saying (quoted from the Grundzuge), "we want to admit the naive notion of set, but observing the restrictions which cut off the way to that [the--my comment] paradox[es]." The works of Skolem, Esser, C.C. Chang, Hinnion, Brady, P.C. Gilmore, Lirbert, and others still researching Naive Set Theory come to mind.... $\endgroup$ Jan 27 '13 at 13:48
  • $\begingroup$ @John: 'Lirbert' should be "Thierry Libert". Sorry. $\endgroup$ Jan 27 '13 at 13:52
  • $\begingroup$ @Andres: Since ZF via Separation and Replacement is infinitely axiomatizable, any finite (or infinite) list of axioms derived from the Separation and Replacement schema are fragments and might be deemed falling short of the mark (I hope that this is not too silly or stupid a remark--if it is, my apologies...). Could, for example, one show that large cardinal axioms could not be construed as instances of replacement or separation (I'm thinking of Harvey Friedman's paper, "The Axiomatization of Set Theory by Extensionality, Separation, and Replacement")? Is this just a silly idea? $\endgroup$ Jan 27 '13 at 14:27
  • $\begingroup$ The actual title of the Friedman paper is "The Axiomatization of Set Theory by Extensionality, Separation, and Reducibility". Sorry. $\endgroup$ Jan 27 '13 at 15:53

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