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November 17, 2021

Large Sets: The Movie

Posted by Tom Leinster

Earlier this year, I wrote a series of blog posts on large sets — or large cardinals, if you prefer — in categorical set theory. Thinking about large sets in Glasgow’s beautiful green spaces, writing those posts, and chatting about them with people here at the Café was one of the highlights of my summer.

Juan Orendain at the Universidad Nacional Autónoma de México was kind enough to invite me to give a talk in their category theory seminar, which I did today. I chose to speak about large sets, first giving a short introduction to categorical set theory, and then explaining some of the key points from this summer’s blog posts.

You can watch the video or read the slides.

Posted at November 17, 2021 10:48 PM UTC

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Re: Large Sets: The Movie

How would one define a Reinhardt set (i.e. Reinhardt cardinal in material set theory) in structural set theory (possibly with universes), and prove in a structural manner that the existence of a Reinhardt set is inconsistent with ETCS+Replacement?

Posted by: Madeleine Birchfield on November 20, 2021 4:09 AM | Permalink | Reply to this

Re: Large Sets: The Movie

That’s hard. The way I would try (if I was convinced it was possible and I just had to do it) is to make the model of AST from ETCS+R, then somehow use the ZF-algebra structure to figure out what a rank-into-rank embedding looks like. And then…. ?

Posted by: David Roberts on November 20, 2021 11:33 AM | Permalink | Reply to this

Re: Large Sets: The Movie

Because no one else has said it, I just want to say that I really enjoyed your talk (I saw it via the YouTube recording). And I agree more and more that “beths exist” is a really nice axiom. It’s overkill, still, for Borel Determinacy, but it doesn’t suffer from the reasonably arbitrary cut-off height than the sharp amount of powersets needs (countable-ordinal-many, for all countable ordinals). Very Cantorian, as has been pointed out elsewhere.

Posted by: David Roberts on November 20, 2021 11:37 AM | Permalink | Reply to this

Re: Large Sets: The Movie

Thanks, David! That’s kind of you to say so.

Posted by: Tom Leinster on November 22, 2021 11:34 AM | Permalink | Reply to this

Leinster/Shulman terminology

Thanks, Tom, very much for giving such a clear, succinct summary of the situation.
The following is probably well-known, but maybe it would be useful to address it again here, or perhaps provide a reference for it:
To what extent are the phrases on the left and right of the following table synonymous?

(1)Leinster Shulman traditional, membership-based set theory material set theory categorical set theory structural set theory \boxed{ \begin{array} {c|c} \text{Leinster} & \text{Shulman} \\ \text{traditional, membership-based set theory} & \text{material set theory} \\ \text{categorical set theory} & \text{structural set theory} \\ \end{array} }

For Mike’s terminology, I am referring in particular to two of his works:
From Set Theory to Type Theory, 2013 nCafé
Comparing material and structural set theories, 2018 arXiv

Posted by: Keith Harbaugh on November 22, 2021 1:26 AM | Permalink | Reply to this

Re: Leinster/Shulman terminology

I really should add that the second Shulman reference, “Comparing material and structural set theories”, does much more than merely setting terminology, but goes far to address the comparison issues.

Posted by: Keith Harbaugh on November 22, 2021 12:12 PM | Permalink | Reply to this

Re: Leinster/Shulman terminology


It’s a good question.

First, I tend to say things like “traditional, membership-based set theory” rather than the shorter “material set theory” just so as to avoid having to explain what “material” means. I’m not aware of any real difference between the two terms, although there may be some subtleties.

(I remember a well-known material set theorist saying that they didn’t like the term material set theory, because it was only ever used by people who are critical of the subject. That’s probably true.)

Second, I’d say that “categorical set theory” is more specific than “structural set theory”. Mike’s theory SEAR (Sets, Elements, And Relations) is given on the nLab as an example of a set theory that is structural but not categorical. Though maybe it doesn’t quite fit the letter of the definition of “structural set theory” in Mike’s “Comparing material and structural set theories”, which says:

we will use the term […] structural set theory for theories which take functions as fundamental,

whereas SEAR doesn’t literally take functions as fundamental. (It takes relations as fundamental, and a function is a special relation.)

As well as “structural” and “categorical”, there’s “isomorphism-invariant”. And then there’s the not wholly developed notion of neutrality.

Honestly, I probably use these terms more interchangeably than I should, especially when I’m giving a talk or writing something informal on a blog.

Posted by: Tom Leinster on November 22, 2021 12:28 PM | Permalink | Reply to this

Re: Leinster/Shulman terminology

As for me, now that I’m mostly into type theory instead, I don’t care so much as I used to about having snappy names for the different kinds of set theory. It would have been nice if “material” had taken off, but nowadays I probably use “membership-based” at least as often myself, for the same reasons. I agree with Tom that they’re basically synonymous.

I also agree that the situation with “structural” and “categorical” is somewhat murkier. I do consider SEAR to be a structural set theory, and I agree that that does seem to contradict the passage Tom quoted from my 2018 paper. However, in that case, the paper even contradicts itself, since a couple of paragraphs later I wrote

Structural set theories … are usually formulated in the language of category theory, [but] there is nothing intrinsically necessary about that (see e.g. [S+09]).

with the citation being to SEAR! (True, the passage Tom quoted was carried over mostly verbatim from the ancestor 2010 paper Stack semantics and the comparison of material and structural set theories — but that’s no excuse, since SEAR dates back to 2009.)

If SEAR is structural but not categorical, then what is the defining feature of a categorical set theory? After all, the sets and functions in any kind of set theory do form a category. Do we require that a “categorical set theory” include the axioms of a category explicitly among its own axioms? If so, then “being categorical” would really be a property of a presentation of a theory.

And, of course, sets and relations also form a category — although the axioms of SEAR don’t explicitly assert this fact, but derive it from a general principle of relational comprehension. I bet one could write down a presentation of ETCS that doesn’t include associativity of composition explicitly, and maybe not even function composition explicitly, instead deriving them from a principle of function comprehension. Would that no longer be a categorical set theory?

Ultimately there’s probably not much point in splitting hairs like this. Clearly ETCS and SEAR belong to the same general circle of ideas and style of theories, at least when compared to ZFC. Sometimes I say things like “structural/categorical set theory” to try to get the advantages of both words.

Posted by: Mike Shulman on November 24, 2021 5:33 AM | Permalink | Reply to this

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