Rethinking set theory · HackerPapers discussions

Rethinking set theory — mentioned by griffzhowl

Wed, 01 Oct 2025 15:40:15 GMT

> Category theory is what you get when you take mappings instead of sets as the primitive objects of your universe. I'm not sure about that, because you still need some concept of set (or collection or class) to define a category, because you need a set of objects and mappings between them (technically that's a "small" category, but to define any larger category would require at least as much set-theoretical complication). More exactly, whereas in set theory it's the membership relation between sets and their elements that is basic, in category theory it's the mapping between objects.…

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Rethinking set theory — mentioned by ogogmad

Tue, 30 Sep 2025 18:45:53 GMT

It might be relevant to look at this: https://home.sandiego.edu/~shulman/papers/jmm2022-complement... Also this: https://arxiv.org/pdf/1212.6543 Assuming you haven't looked at these already, of course.

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Rethinking set theory — mentioned by Twisol

Tue, 15 Nov 2022 01:37:24 GMT

Category theory might be easier to think about as a different way to think about the same objects we've always studied in mathematics. Since it's a formal theory (just like set theory), we can also study category theory mathematically. In fact, we can study set theory categorically and we can study categories set-theoretically -- they aren't really at odds with each other. If you wanted to study set theory from a categorical perspective, you would likely be interested in topos theory, which captures essential properties of set theory and generalizes them to a variety of interesting settings.…

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Rethinking set theory — mentioned by Twisol

Wed, 02 Sep 2020 17:55:22 GMT

> From what I've learned so far, the set theory suffers from Russel's Paradox So-called "naive set theory" does, but my understanding is that this paradox (and others like it) sparked a crisis in mathematics that led to the creation of ZFC and others. ZFC is not known to be inconsistent (due to deep and technical reasons, I can't claim positively that it is consistent), and it was designed to avoid the paradoxes that plagued naive set theory. Russell's type theory was another early contender for a formalism. For whatever reason (I'm not aware of all the details), Zermelo-Fraenkel set theory…

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Rethinking set theory — mentioned by Kednicma

Fri, 14 Aug 2020 06:31:08 GMT

I'm obliged, I suppose, to list off a few big wins. From the 50s and 60s, we have classic theorems which use abstract nonsense to generalize big statements about entire classes of objects, like Freyd's adjoint functor theorem [0], Yoneda's lemma [1], and Lawvere's fixed-point theorem [2]. Yoneda's lemma is the slogan that "an object is equivalent to the arrows coming/leaving it", but formal. Lawvere's theorem is a deep and permanent generalization of Gödel, Tarski, Turing, et al. on incompleteness and undecideability. Starting in the 60s and continuing to the present, there's been a theme of…

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Rethinking set theory — mentioned by triska

Wed, 18 Dec 2019 21:41:59 GMT

These are excellent questions! And yes, there is! For example, I highly recommend the paper Rethinking set theory by Tom Leinster: https://arxiv.org/abs/1212.6543 It highlights Lawvere set theory. The paper won this year's Chauvenet Prize: https://www.maa.org/programs-and-communities/member-communit...

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Rethinking set theory — HN submission

Sun, 25 Aug 2019 13:02:57 GMT

Rethinking Set Theory (2012)

3 points; 0 discussion replies on HN. Paper and source history. A submission is not an endorsement.

Rethinking set theory — mentioned by ocfnash

Mon, 03 Jun 2019 10:49:39 GMT

A substantial portion of this text appears to be concerned with traditional Zermelo–Fraenkel set theory (and its extensions). I have gradually come to believe that ZF theory has received a disproportionate amount of attention on account of the fact that it serves as the "official" foundations of mathematics, but that it is not an especially beautiful, or useful, structure. I believe that ZF theory is an interesting object, worthy of mathematical study, but _not_ the best candidate for the foundations of mathematics! I am very happy to see that this year's Chauvenet Prize [1] was won by Tom…

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