Category talk:Set theory

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I don't entirely agree with the proffered distinction among different meanings of "set theory", which I think comes from a formalist POV. Perhaps we can negotiate wording that acknowledges the realist POV without branding it "naive".

Here's the thing: The antinomies (e.g. Russell) all basically result, not from Cantor's notion of a set as a "collection into a whole [...] of definite and separate objects [...] of our intuition or thought", but rather from Frege's notion of set as an extension of a (presumably definable) property. Russell's paradox was a serious (I would say mortal) blow to Frege's conception, but not at all to Cantor's, which I think Zermelo (surely the father of "axiomatic set theory") followed quite closely. Certainly Zermelo did not consider a set to be "anything that satisfies the axioms"; this is very clear from his considerations on second-order logic.

The notion that all models of the ZFC axioms have equal standing is an extreme minority view among working set theorists (for good reason, BTW), and I don't think it should be presented unchallenged on this page. --Trovatore 28 June 2005 06:42 (UTC)

I may be in part responsible for that language being included, and I certainly do not sympathize with formalism. The word "naive" I was follows conventional terminology popularized by Paul Halmos. But in one sense it is "naive": it's how ordinary mathematicians, as opposed to set theorists, think, it seems to me. Michael Hardy 28 June 2005 20:06 (UTC)

Certainly Zermelo did not consider a set to be "anything that satisfies the axioms";

I wouldn't want to claim that he did think of a set as "anything that satisfies the axioms", but only that that seemed to be what axiomatic set theory seems to have become. Michael Hardy 28 June 2005 20:06 (UTC)

The notion that all models of the ZFC axioms have equal standing is an extreme minority view among working set theorists (for good reason, BTW),

Oh -- that's what you meant. I agree with you on this point. I meant that in a certain technical sense, the axiom system treats them as equal. Nonetheless, one prefers one model over another because of one's "naive" (i.e., intuitive or philosophical) point of view. Michael Hardy 28 June 2005 20:06 (UTC)

That's fine, but the page remains problematic. The natural reading is that workers in axiomatic set theory think of sets as just whatever happens to satisfy set-theoretic axiom systems, and that's just not true. --Trovatore 29 June 2005 03:17 (UTC)

list of set theory topics

Could others help expand the list of set theory topics? I know there are some people who think categories can or should supersede topics lists, but those people have been resoundingly and deservedly rebuffed whenever they've tried to list a topics list on votes for deletion for that reason. Topics lists are an enormously superior thing for a number of reasons, among them: (1) they can be organized (see list of geometry topics, list of combinatorics topics, list of probability topics); topics lists obviously cannot be organized; (2) they can be moved if their title is misspelled or in any way could be improved. Michael Hardy 28 June 2005 20:02 (UTC)

I can think of lots of things to add. Is it OK if they're red links for a while? On another note--the list is pretty haphazard at the moment. Should topics be alphabetized? Made more hierarchical? --Trovatore 29 June 2005 03:17 (UTC)

Descriptive set theory category

Currently there's one very meager article on Descriptive set theory, but in my opinion it really deserves its own category. I could easily list twenty good topics for articles to include, most of them not written yet, of course. But I'm a little foggy on a couple of points:

• Can a category have the same name as an existing article? Or could the article be moved to a category page?
• I'd like to make "Descriptive set theory" a subcategory of "Set theory", but I haven't found any easy way to navigate up the DAG to parent categories. Is it OK to mark an article both as category A and category B, when B is a subcategory of A?

--Trovatore 29 June 2005 03:30 (UTC)

Category:Mathematics vs. Category:Mathematical logic

Category:Mathematical logic is itself a category within Category:Mathematics. — Robert Greer (talk) 13:42, 13 March 2009 (UTC)