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I am curious as to how much of Mathematics can be derived using Naive Set Theory. I know for example, probability can be developed from Naive Set Theory. How much more of mathematics can be developed from Naive Set Theory?

Thanks!

Andrés E. Caicedo
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Corylikesmath
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  • Probability will need measure theory at some point, and thus one enters non-naive set theory (which is not really a very well-deined term). But to understand measure theory well, can require serious set theory. – Henno Brandsma Mar 07 '17 at 13:55
  • Since Naive Set Theory isn't even consistent, I'd think to say it doesn't develop any part of mathematics. – Hayden Mar 07 '17 at 13:56
  • @Hayden, as I mention in my answer these inconsistencies are somewhat academic as far as the practice of traditional mathematics is concerned. In fact most practicing mathematicians outside of set theory do not know formal set theory. – Mikhail Katz Mar 07 '17 at 13:58
  • I don't think this question is well-posed: it is not at all clear what it would mean for some mathematics to be "developed from naive set theory". – tomasz Mar 07 '17 at 14:06
  • @tomasz, the kind of definitions that are usually written down in "ordinary" mathematics (whatever that means) just don't run into paradoxes that emerged around 1900 that led to the development of axiomatic set theory. – Mikhail Katz Mar 07 '17 at 14:11
  • @MikhailKatz: Of course. If they did, they would be contradictory. But I don't see how it relates to what I said about the question being ill-posed. – tomasz Mar 07 '17 at 14:13
  • It is my understanding that Naive Set Theory, following basic set operations can be used to develop the ideas of probability - for example, combinatorics. – Corylikesmath Mar 07 '17 at 14:15
  • I don't see what it means for mathematics do be "developed by XXX set theory" in general, I guess. Even less so if XXX is inconsistent. If anything, it would make sense to ask if a theory (set, type, category or whatnot) can interpret a part of mathematics. But I don't see any benefit of such interpretation by something known to be internally inconsistent. Of course, one can use suitably restricted naive set theory for that purpose and that would make sense, but once you get suitably formal with that, it's no longer naive. – tomasz Mar 07 '17 at 14:18
  • @tomasz, the inconsistency you mention only arises through the use of self-referential formulas and the like, as you surely know. So long as such formulas are not used, naive set theory can be a basis, and in fact is a basis for most "ordinary" mathematics since, as I mentioned, most mathematicians outside set theory don't even know formal set theory. Consider, for example, Leibniz's approach to analysis with infinitesimals. This was successfully interpreted by Abraham Robinson three centuries later in 1961,... – Mikhail Katz Mar 07 '17 at 14:23
  • ... but surely you would admit that Leibnizian mathematics did not undergo a jump from something totally worthless before 1961, to a tremendously important field a moment after Robinson's paper was published in a refereed journal. – Mikhail Katz Mar 07 '17 at 14:23
  • Not really sure why this got downvoted - I think it's a good question. – Noah Schweber Mar 07 '17 at 14:29
  • @MikhailKatz: I'm not claiming anything of the sort. I don't think any jump of that sort happened around the turn of twentieth century, either. As far as I know, there was no concept of a set, naive or not, during Leibniz's time, and my point stands: I don't believe set theory, naive or otherwise, develops any mathematics (except perhaps set theory itself). It can be used (as a tool) in some parts, and it may be convenient in formalisations, but that's about the extent of its relevance. (I don't mean to say that it is a small extent, though, don't get me wrong.) – tomasz Mar 07 '17 at 14:35
  • @tomasz, I am not sure your remarks are historically accurate. I believe there were indeed naive notions of set already in the 17th century. As you know both Galileo and Leibniz struggled with the paradoxes that a proper subset can be equinumerous with the set itself. There is considerable historical literature on this sort of discussion. – Mikhail Katz Mar 07 '17 at 14:39
  • @Corylikesmath, you can vote to re-open the question by clicking on the button just below your question. – Mikhail Katz Mar 08 '17 at 08:24

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As argued in this recent comment "most" mathematics does not require a formal set-theoretic foundation. Here the meaning of "most" is debatable; certainly modern set theory cannot function in the context of naive set theory. However the paradoxes that were eventually shown to plague naive set theory don't affect the practice of the traditional fields of mathematics like analysis, geometry, algebra, topology (surely I offended somebody by not including his field and I apologize in advance; I merely listed fields most relevant to my own research).

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Mikhail Katz
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  • So then I could say that Naive Set theory can be used to develop analysis, geometry, algebra, topology, etc? – Corylikesmath Mar 07 '17 at 14:01
  • Well, this should be qualified by "most" certainly, if only to account for the fact that sometimes it happens that rather intricate results in set theory do have applications in "ordinary" mathematics. A case in point is the existence of large cardinals which at least on the face of it was used in the construction of big machines that go into the proof of Fermat's last theorem. Later people argued that such assumptions are not really needed, but as far as I know there is no definitive text on this. – Mikhail Katz Mar 07 '17 at 14:04
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    @Corylikesmath Here's a couple other examples of non-naive set theory being used in mathematical practice. First, large cardinals were used in a crucial way to provide the original proofs of some results about left distributive algebras; this is stronger than the FLT case, since (my understanding is) it was clear from the get-go in FLT how to get rid of the large cardinals. And there are other results around left distributive algebras for which no proofs without large cardinals are known. Meanwhile, Harvey Friedman has looked at combinatorial principles requiring large cardinals, (cont'd) – Noah Schweber Mar 07 '17 at 14:14
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    and questions about regularity properties (measurability, property of Baire, etc.) about complicated-but-still-definable types of set (e.g. start with the Borel sets and close off under continuous image and complement) are independent of ZFC, and require large cardinals for affirmative answers (this is studied in descriptive set theory). (On FLT see, the discussion here, and for left distributive algebras see here - – Noah Schweber Mar 07 '17 at 14:20
  • @NoahSchweber, thanks for some nice examples. – Mikhail Katz Mar 07 '17 at 14:24
  • for Friedman's stuff see here, and for large cardinals in descriptive set theory see this lovely paper by Larson. (And +1 to the answer btw.) By the way, none of this is to dispute Prof. Katz' answer that most of mathematics does not require non-naive set theory; I agree with that statement, and have argued for it elsewhere on this site. – Noah Schweber Mar 07 '17 at 14:27
  • @NoahSchweber, I think this close is questionable and voted to re-open. – Mikhail Katz Mar 08 '17 at 08:27