The n-Category Café

Regarding Dedekind, he wrote down a proof of the result without choice while in the later stages of drafting Was sind und was sollen die Zahlen? (in July 1887), but didn’t include it in that monograph. However, he did include a related result which essentially gives it to you, using a very similar method of proof - and it is (I think) the only statement not explicitly proved in that text, and left to the reader. As you say, this was in the middle of the period with a strained relationship with Cantor.

For those who are interested, Arie Hinkis’ book Proofs of the Cantor-Bernstein Theorem digs into the history.

Wow, David, that’s an amazingly comprehensive book! What a labour of love. Thanks for linking to it; I had no idea it existed.

It’s also free through the link David gives.

I’d like to sit in your class and hear you tell that story about the envoys!

I don’t deny that there’s a lot of categorical thinking going on in this course on sets (even though I don’t mention categories), just as there may be a lot of ring-theoretic thinking going on in a course on number theory (even if they never mention rings).

I hope the students will find that a good deal of what they learn is useful for other subjects too. It’s not my primary purpose to teach them generally useful stuff, but I do think it’s a substantial fringe benefit of a categorical approach (even a secretly categorical approach) that you learn more broadly applicable concepts than you do in a ZFC-based course.

However, with each week that passes, the course does feel more particular to sets and less about general categorical notions. For example, last week’s proof of Cantor’s theorem on $X \lt P(X)$ and the Cantor–Bernstein theorem are highly specific to sets. General categorical thinking won’t help you much there.

What we’re doing this week (in notes I’ll release soon) is the definition and structure of $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$. The material on $\mathbb{N}$ inevitably starts off by using the axiom that there exists a natural numbers object. But pretty soon in the development, everything starts looking just about the same as in any other axiomatization of set theory.

The topics for the remaining weeks are well ordered sets, the axiom of choice, and cardinal arithmetic. This will be really very close to the development that you’d see in a ZFC-based course. The main difference is that everything will be isomorphism-invariant — so, no transitive sets etc. But the results are essentially all the same.

I also wonder if the amount of category theory is actually necessary for ETCS and I suspect it mostly is.

I’ve tried to keep the generalities to a minimum. That doesn’t mean I’ve succeeded: I’m not naturally a concise writer, and these notes are a first iteration written under great time pressure. Plus, while I did detailed advance planning of the whole course, I’m not one hundred percent sure which results I have to include in earlier chapters in order to be able to use them in later chapters. So there’s probably some fat that could be trimmed, despite my best efforts.

Dr. Kirby

You write:

“I also wonder if the amount of category theory is actually necessary for ETCS and I suspect it mostly is.”

There is a reason for this which is obscured in “a battle of set theories” (Dr. Shulman’s material/structural distinction). The debates which had occurred on the FOM mailing list between Dr. Friedman and category theorists had been about advocacy over the first-order paradigm — not “sets.” When one reads Lawvere and Rosebrugh, the significant element of category diagrams for reconstruction of “set intuition” are inclusions. Historically, one may compare this with more modern accounts of the mereological part relations.

Where Lawvere and Rosebrugh elevate Dedekind and criticize philosophers, they ignore Russell’s critique of Dedekind relative to Peano. Russell’s observations have been restated in “Set Theory and Its Philosophy” by Michael Potter.

In effect, inclusions approach “individuation” via singleton sets. This is how Zermelo had approached the idea in his 1908 paper (If I recall correctly, it also appears in Padoa’s paper where models for identifying axiom independence is introduced.). For Zermelo, this use of singleton sets appears motivated by a need to accommodate urelements. However, the logicist view of set theory had changed to an algebraic view with Skolem’s criticism of “definite properties” in Zermelo’s paper. First-order advocates will always object to conflation of “an individual” with the singleton having an individual as an element. And, this goes back to Russell’s comparison of Dedekind and Peano.

With Skolem, support for urelements becomes an axiom for a “set” of urelements (everything is a set) unnecessary for mathematics (a theory of pure sets suffices).

If one understands Lawvere’s contribution as a reverse engineering of symbolic logic, then one can have a deep appreciation for how “element” and “member” are interpreted by Lawvere. Along similar lines, where the status of “names” is central to the linguistic paraphrases of symbolic logic, the explanation of “names” in McClarty is equally impressive.

Once one acknowledges that the debate involves individuation and interpretation for the sign of equality, the two papers to compare are Lawvere’s paper on the unity of opposites

https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1996-unity-and-identity-of-opposites-in-calculus-and-physics.pdf

and Russell’s rejection of “differences” on PDF page 7 of his paper “On Denoting,”

https://www.philosophy-index.com/russell/on-denoting/Russell-On_Denoting.pdf

For what this is worth, Russell’s view is useful to the verificationist perspective of homotopy type theory. It is implicit to the sequent axiomatization found in the guest post on identity types sponsored by Dr. Riehl. And there is a particularly unfortunate circumstance that makes this so difficult to sort out — logicism, first-order logic, and Herbrand logic share the same inference rules. So, when one portrays the situation as “a battle of set theories,” it is difficult to figure out who is arguing against what.

“The more I read of your notes, the more I wonder if it is really a course in category theory or in set theory.”

It’s a course in set theory. A course on category theory will inevitably talk about functors and adjunctions and natural transformations and general limits/colimits and the Yoneda lemma and those are completely missing from this course.