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arXiv:1212.6543math.LO

Rethinking set theory

Tom Leinster

Abstract

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.

arXiv comments
8 pages, expository
Journal reference
American Mathematical Monthly 121 (2014), no. 5, 403-415
DOI
Subjects
math.LOLogic math.CTCategory Theory

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Linked by 6 identifiable commenters in 7 comments across 7 threads, from Jun 2019 to Oct 2025. Found in 1 HN submission from Aug 2019 to Aug 2019. Open submission discussions.

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Discussed in comments7 threads

  1. Category Theory Illustrated – Natural Transformations 1 comment

    • > 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…

      griffzhowlView on HNParent context

  2. How has mathematics gotten so abstract? 1 comment

    • 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.

      ogogmadView on HNParent context

  3. Topology, a Categorical Approach (2020) 1 comment

    • 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…

      TwisolView on HNParent context

  4. Programming with Categories 1 comment

    • > 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…

      TwisolView on HNParent context

  5. Kindergarten Quantum Mechanics 1 comment

    • 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…

      KednicmaView on HNParent context

  6. List of Statements Independent of ZFC 1 comment

    • 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…

      triskaView on HNParent context

  7. Foundations of Mathematics (2015) [pdf] 1 comment

    • 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…

      ocfnashView on HNParent context

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