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Madeleine Birchfield (Aug 14 2024 at 15:30):Split off from this thread
Is it possible for mathematics to undergo Kuhnian revolutions in a similar way to how science undergoes Kuhnian revolutions?
John Baez (Aug 14 2024 at 15:36):The hard part for most of us may be figuring out (or learning) what's a "Kuhnian revolution".
Jonas Frey (Aug 14 2024 at 15:45):I always thought that the title of Benabou's "Fibered categories and the foundations of naive category theory" was a nod towards Kuhn, which I rembered as distinguishing "naive", "formal", and "critical" phases of a theory. But when I asked on philosophy stackexchange , I was told that these terms might be rather associated with Lakatos.
But maybe I should have rather asked here! Does anybody know what Benabou might have been referring to?
John Baez (Aug 14 2024 at 16:32):It reminds me of Halmos' famous book Naive Set Theory.
John Baez (Aug 14 2024 at 18:11):That book came out in 1960. But I don't know if that's the first use of the term "naive set theory" - the preface might say something.
Jonas Frey (Aug 14 2024 at 20:38):John Baez said:
It reminds me of Halmos' famous book Naive Set Theory.
Ahh, probably that's what Bénabou was referencing.
Jencel Panic (Aug 15 2024 at 06:36):AFAIK naive set theory is the one in which there are no restrictions on the way sets are formed (i.e. the one susceptible to Russell's paradox)
Jencel Panic (Aug 15 2024 at 06:38):From the wiki page of "Naive Set Theory":
While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of ZFC set theory
Jencel Panic (Aug 15 2024 at 06:43):In CS we have the term "naive solution" meaning "something that is kinda correct, but too simple to really work (e.g. takes too much time and/or space)."
https://cs.stackexchange.com/questions/33914/what-is-a-naive-method
David Michael Roberts (Aug 15 2024 at 07:05):I would say the technical naïveté in Halmos' book is down to his axiom of unions talking about taking the union of "every collection" of sets. He doesn't explain what he means by a "collection", and this is clearly false if one takes "collection" to mean "class". But this is fairly surface-level, since one could say "collection" means "set" and then it's fine.
I think the WP page is not correct since the book doesn't introduce the schema of Replacement. It seems to me that (modulo the statement axiom of unions) it actually gives the axioms of Zermelo set theory.
David Michael Roberts (Aug 15 2024 at 13:45):Actually, I take back my complaint about reference "collections". In the first section, Halmos specifies that he uses "collection" as a synonym for "set" for the sake of stylistic variation. Certainly it helps to say "collection of sets" rather than "set of sets" in this introductory text, while at the same time maintaining the underpinning theory.
David Michael Roberts (Aug 15 2024 at 14:37):And I have to take back my claim he doesn't give Replacement. It's in what seems like an odd place, much later in the book. And it's held off until it's needed to define the (infinite) von Neumann ordinals.
Jonas Frey (Aug 15 2024 at 17:21):David Michael Roberts said:
And I have to take back my claim he doesn't give Replacement. It's in what seems like an odd place, much later in the book. And it's held off until it's needed to define the (infinite) von Neumann ordinals.
Ahh I just had a quick look myself and didn't find it! Can you tell me where?
David Michael Roberts (Aug 15 2024 at 21:12):@Jonas Frey He calls it the axiom of substitution and it's in the chapter on ordinal numbers. And I figured out it's actually the Collection schema, not Replacement.
David Michael Roberts (Aug 15 2024 at 21:29):The statement is roughly
if S(a, b) is a sentence such that for each a in a set A the set B = {b: S(a, b)} can be formed, then there exists a function F with domain A such that F(a) = {b: S(a, b)}.
In fancy language it's saying that if you have a relation from a set A to V that is 'small' (or maybe in fact its opposite is small) in the sense of algebraic set theory, then there is a function f: A->B (B is a set, in Halmos' definition) whose value f(a) at a in A is the set of sets related to a by the class relation.
David Michael Roberts (Aug 16 2024 at 13:03):The WP page is now updated https://en.wikipedia.org/wiki/Naive_Set_Theory_(book) to explain the axiom system Halmos gives.