Tom Leinster
2012 · arXiv
Mathematicians manipulate sets with confidence almost every day, rarely making mistakes. Few of us, however, could accurately quote what are often referred to as "the" axioms of set theory. This suggests that we all carry around with us, perhaps subconsciously, a reliable body of operating principles for manipulating sets. What if we were to take some of those principles and adopt them as our axioms instead? The message of this article is that this can be done, in a simple, practical way (due to Lawvere). The resulting axioms are ten thoroughly mundane statements about sets. This is an expository article for a general mathematical readership.
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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...
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 Leinster's "Rethinking set theory", in which he highlights that Lawvere set theory…
looks like a much better candidate. I cannot do better than recommend you look at Leinster's superb article. [1] MAA, Chauvenet Prize, https://www.maa.org/programs-and-communities/member-communit... [2] Leinster, T., "Rethinking set theory", https://arxiv.org/abs/1212.6543
> 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. Nevertheless, the basic concepts of set theory can also be defined within category…
theory, so in that sense they're inter-translatable. In each case though, you need some ambient idea of a collection (or class or set) of the basic objects. Tom Leinster has a brilliantly clear and succinct (8 pages) exposition of how this is done here https://arxiv.org/abs/1212.6543 The thing is, even defining first-order logic requires a (potentially infinite) collection of variables and constant terms; and set theory is embedded in first-order logic, so both set theory and category theory are on the same footing in seemingly requiring a prior conception of some kind of potentially infinite "collection". To be honest I'm a bit puzzled as to how that works logically
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.
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. There's also a really good (and IMHO approachable) paper by Tom Leinster [1] on…
how to see set theory from a categorical perspective, without going all the way out to toposes. I think it's a good way to understand what category theorists try to emphasize. If you wanted to study categories from a set-theoretic perspective, you're in luck -- that's probably the most common approach to learning category theory in the first place! Studying categories with categories is also possible, but you already need to be fairly comfortable with categories in order to model them internally. (Studying set theory with set theory is much the same -- models of ZFC can get wild.) (You can also learn category theory from first principles, without thinking in terms of sets. This isn't really any different from set theory, either -- set theory is often peoples' first introductions to abstract mathematics at the undergraduate level, so there isn't really much choice in the matter. For my money, though, I think category theory from first principles is a little less weird.) [1]: https://arxiv.org/abs/1212.6543
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 exploring category theory as a prime foundation for maths. There's been complete…
foundations like the Elementary Theory of the Category of Categories (ETCC) [3], and also much smaller but pointed presentations that focus on just simplifying set theory. I like [4] in particular. On a deeper level, in terms of structure and philosophy, the entire existence of homotopy type theory (HoTT) relies on categorical presentations and memes. HoTT is, more than any other type theory, an investigation into what equivalence means, and it dovetails wonderfully with 2-category theory and the understanding that algebraic laws can be transformed into transformations. [0] https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_fu... [1] https://en.wikipedia.org/wiki/Yoneda_lemma [2] http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf [3] https://ncatlab.org/nlab/show/ETCC [4] https://arxiv.org/abs/1212.6543
> 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 (later extended with the Axiom of Choice) won the popular mindshare. Personally,…
I'm more a fan of the Elementary Theory of the Category of Sets (ETCS) [1], which is indeed drawn from category theory. But it's equivalent in deductive power to ZFC, so what you pick mostly only matters if you're doing reverse mathematics. (This is what really answers your question, I think.) Russell's type theory and modern type theories are distinct (and there is no single "type theory"). I'm led to believe the commonality is primarily with the usage of a hierarchy of universes, so that entities in one unverse can only refer to entities in a lower universe. [1] "Rethinking set theory", Tom Leinster: https://arxiv.org/abs/1212.6543
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