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:
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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').
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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?
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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?
Best Answer
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.