Timeline for Hausdorff and Naive Set Theory
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Nov 27, 2023 at 17:58 | vote | accept | Thomas Benjamin | ||
| Nov 25, 2023 at 14:04 | comment | added | Paul Taylor | "his book on set theory (the English translation)". Beware that the English translation was of a much later German edition (and in particular omits the point-set topology from the original). Free scans of the 1914 book can be found online, although I forget where. | |
| Jan 26, 2013 at 2:37 | answer | added | John Stillwell | timeline score: 10 | |
| Apr 15, 2012 at 21:57 | comment | added | Thomas Benjamin | I am also considering asking the question you suggest either by editing my question to suit or asking it as a new question. Perhaps questions 2 and 3 would be better suited to philosophy.stack.exchange? | |
| Apr 15, 2012 at 21:50 | comment | added | Thomas Benjamin | @quid: so you are basically saying Hausdorff considered 'Set Theory' not as a 'theory proper' but merely as a language in which mathematics can be formulated? If that is the case then it would seem that Unrestricted Comprehension could be construed not as an axiom but as a rule of formation for well-formed formulas and the 'paradoxical sets' as grammatical but meaningless sentences (formulas) like "Colorless green ideas sleep furiously." Is this essentially what you are saying? | |
| Apr 15, 2012 at 11:26 | comment | added | user9072 | ...as I am convinced they merely come from reading way to much into some vague comment. My suggestion would be to edit the wuestion to reduce it to something like: I read that H. considered attempts of some of his contemporaries at axiomatization of ST as premature. Now with a hundred years of hindsight, does it seem he was rather right or wrong. I could imagine that some here have somthing interesting to say on this. I don't know, say, I think some people could think one might have better chosen a weaker AC to have less pathologies in analysis, while others disagree. Or, something else. | |
| Apr 15, 2012 at 11:19 | comment | added | user9072 | ...what other problems come up. So, that not the 'solution' to the first five makes the solutions of the 20th to 30th even more difficult. So H. might (I don't know) just have thought something like: we do not understand all this well enough yet, let's explore it informaally/naively a bit more to see what we actually need in the end. All that being said, I think your question (1) is interesting and also MO seems suitable to ask it (elbeit it is a bit subjective). By contrast, (2) and (3) IMHO, for the reasons explained above, are not real questions... | |
| Apr 15, 2012 at 11:14 | comment | added | user9072 | that there is a cricle of ideas that seems fruitful, but it is not yet developped enough to be considered (by some) as mathematics 'proper'. Then you describe this state of affairs 'somehow'. I also disagree that H. necessarily had something specific to say on the paradoxes to be legitimized to say trying to resolve them via axiomatization is premature. If you try to develop something (a machine, a programm, a theory) and the first problems come up, some might hold the opinion one has to address them right away, some might say it is wiser to proceed in informal mode and wait ... | |
| Apr 15, 2012 at 11:06 | comment | added | user9072 | The way I read the quote of Scholz, it is not that he [Scholz] "brought semiotics into the mix" (because this is in H. work) but S. is merely quoting how H. viewed set theory: as a 'semiotic tool', to do mathematics (in the sense what he at that time consider as such), as opposed to viewing set theory (already) as a (new) subfield of mathematics in its own right. That he used the word 'semiotic tool', well, couple of decades latter Grothendieck referred to some things as 'yoga' (motives, anabelian geoemetry, possibly other things). Sometimes in the developpment of math it happens... | |
| Apr 15, 2012 at 7:08 | comment | added | Thomas Benjamin | one's set-theoretic intuitions by continuing to work in Naive Set Theory with the paradoxes intact. Perhaps I will take Spice's advice and ask this on philosophy.stack.exchange. | |
| Apr 15, 2012 at 7:02 | comment | added | Thomas Benjamin | 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' | |
| Apr 15, 2012 at 6:44 | comment | added | Thomas Benjamin | 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. | |
| Apr 14, 2012 at 23:14 | comment | added | Spice the Bird | 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. | |
| Apr 14, 2012 at 11:57 | comment | added | user9072 | 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.) | |
| Apr 14, 2012 at 10:51 | comment | added | Gerhard Paseman | 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 | |
| Apr 14, 2012 at 10:12 | history | edited | Thomas Benjamin | CC BY-SA 3.0 |
corrected spelling
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| Apr 14, 2012 at 10:07 | history | asked | Thomas Benjamin | CC BY-SA 3.0 |