Timeline for Positive results coming from paradoxes
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5 events
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| Sep 19, 2021 at 9:31 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Dec 16, 2012 at 3:04 | comment | added | Todd Trimble | I would also add that Cantor already knew in essence what came to be called Russell's paradox: he knew, by his eponymous theorem, that no set $V$ could contain its own power set. See en.wikipedia.org/wiki/Cantor%27s_paradox. There's a nice observation at the nLab, at ncatlab.org/nlab/show/Cantor%27s+paradox, that Russell's paradox is the $\beta$-reduced form of the proof of Cantor's theorem as applied to Cantor's paradox. The "target" of Russell's paradox was Frege's Grundgesetze der Arithmetik (if I can put it like that; his letter to Frege was rather kind and gentlemanly). | |
| Dec 16, 2012 at 0:56 | comment | added | Andrej Bauer | Right, I wrote it oversimplified on purpose. The point is that Cantor concieved of set theory, and Russell made the point that things were a bit subtler than they seemed. | |
| Dec 15, 2012 at 15:43 | comment | added | Todd Trimble | This seems like a bit of an oversimplification (I'm sure you're aware of what I'm about to say, Andrej). Certainly there was recognition of the need to restrict the principle of comprehension when Zermelo introduced his axiomatization of set theory, but my understanding is that it was really the discussion of the well-ordering principle and its underpinnings that drove him to carefully specify the axioms of Zermelo set theory with choice. See also Timothy Chow's answer here: mathoverflow.net/questions/28656/… | |
| Dec 15, 2012 at 14:14 | history | answered | Andrej Bauer | CC BY-SA 3.0 |