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July 22, 2021

Large Sets 13

Posted by Tom Leinster

Previously: Part 12.5

This is the last post in the series, and it’s a short summary of everything we’ve done.

Added later: And here’s a talk summarizing it all.

Most posts in this series introduced some notion of large set. Here’s a diagram showing the corresponding existence conditions, with the weakest conditions at the bottom and the strongest at the top:

Diagram of large set conditions

Most of these existence conditions come in two flavours. Take beth fixed points, for instance. We could be interested in the literal existence condition “there is at least one beth fixed point”, or the stronger condition “there are unboundedly many beth fixed points” (meaning that whatever set you choose, there’s a beth fixed point bigger than it).

I’ve been a bit sloppy in using phrases like “large set conditions”. These conditions are as much about shape as size. The point is that if a set XX is measurable, or inaccessible, or whatever, and YY is another set larger than XX, then YY need not be measurable, inaccessible, etc. It usually isn’t. But in ordinary language, we’d expect a thing larger than a large thing to be large.

It’s like this: call a building centennial if the number of storeys is a (nonzero) multiple of 100. Even the smallest centennial building has 100 storeys, so all centennial buildings are what most people would call large. In that sense, being centennial is a largeness condition. But a 101-storey building is not centennial, despite being larger than a centennial building, so the adjective “centennial” doesn’t behave like the ordinary English adjective “large”.

Puzzle   There’s one largeness condition in this series that does behave like this: any set larger than one that’s large in this sense is also large. Well, uncountability and being infinite are both conditions with this property, but it’s something less trivial I’m thinking of. What is it?

In the very first post, I wrote this:

I’ll be using a style of set theory that so far I’ve been calling categorical, and which could also be called isomorphism-invariant or structural, but which I really want to call neutral set theory.

What I mean is this. In cooking, a recipe will sometimes tell you to use a neutral cooking oil. This means an oil whose flavour is barely noticeable, unlike, say, sesame oil or coconut oil or extra virgin olive oil. A neutral cooking oil fades into the background, allowing the other flavours to come through.

The kind of set theory I’ll use here, and the language I’ll use to discuss it, is “neutral” in the sense that it’s the language of the large majority of mathematical publications today, especially in more algebraic areas. It’s the language of structures and substructures and quotients and isomorphisms, the lingua franca of algebra. To most mathematicians, it should just fade into the background, allowing the essential points about sets themselves to come through.

And later on in that post, I continued:

While everyone agrees that the elements of a set XX are in canonical one-to-one correspondence with the functions 1X1 \to X, some people don’t like defining an element of XX to literally be a function 1X1 \to X. That’s OK. True to the spirit of neutral set theory, I’ll never rely on this definition of element. All we’ll need is that uncontroversial one-to-one correspondence.

I’m not sure I did a great job of explaining what I meant by “neutral”, partly because I was still fine-tuning the idea, but also because it’s one of those things that’s easier to demonstrate than describe.

The proof of the pudding is in the eating. These last dozen posts are a demonstration of what I mean by “neutral”. Throughout, sets have been treated in the same way as a modern mathematician treats any other algebraic or geometric object. In particular, everything was isomorphism-invariant, and we didn’t ask funny questions about sets (like “does ξxX\xi \in x \in X imply ξX\xi \in X?”) that we wouldn’t ask in group theory if the sets were equipped with group structures, or in differential geometry if the sets had the structure of manifolds, etc.

As promised, I’ve never needed to insist that the elements of a set XX literally are the functions 1X1 \to X, only that they’re in canonical one-to-one correspondence. Similarly, it’s never been important whether an II-indexed family of sets literally is a function into II, as long as the two notions are equivalent in the appropriate sense.

Aside   Having said that, I don’t want to concede too much of a point. In modern mathematics, it’s absolutely standard that once you’ve established that two kinds of thing are in canonical one-to-one correspondence, you treat them interchangeably. And sometimes you then rearrange the definitions so that there’s just one kind of thing, making the correspondence invisible. The definition of an element as a map from 11 is a case of this.

Personally, when I try to follow arguments that an element should not be defined as a map from 11, I have a hard time finding anything mathematically solid to hold on to. We all agree that the two things are in canonical one-to-one correspondence, so what’s the issue? For comparison, imagine someone arguing that although sequences in a set XX correspond canonically to functions X\mathbb{N} \to X, they shouldn’t be defined as functions X\mathbb{N} \to X. What would you say in response?

I’m mentioning this not in order to restart arguments that many of us have had many times before (though I know I’m running that risk). My point is that although I’ve scrupulously avoided saying that an element is a function out of 11, or that an II-indexed family is a function into II, to do so would actually be pretty neutral in the sense of being mainstream mathematical practice: when two things are equivalent, we treat them as interchangeable.

Let me finish where I started. I began the first post like this:

This is the first of a series of posts on how large cardinals look in categorical set theory.

My primary interest is not actually in large cardinals themselves. What I’m really interested in is exploring the hypothesis that everything in traditional, membership-based set theory that’s relevant to the rest of mathematics can be done smoothly in categorical set theory. I’m not sure this hypothesis is correct (and I suppose no one could ever be sure), which is why I use the words “hypothesis” and “explore”. But I know of no counterexample.

Twelve posts later, I still know of no counterexample. As far as I know, categorical set theory can do everything important that membership-based set theory can do — and not only do it, but do it in a way that’s smooth, natural, and well-motivated.

We could go on with this experiment forever. Personally, I’m very conscious of my own ignorance of set theory. For example, I know almost nothing about two important themes related to large cardinals: critical points of elementary embeddings, and Shelah’s PCF theory. So if the experiment’s going to continue, someone else might have to take the reins.

Posted at July 22, 2021 12:41 PM UTC

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15 Comments & 2 Trackbacks

Re: Large Sets 13

There is a typo in the parenthetical item in the paragraph just below the diagram.

Posted by: jJay Kangel on July 22, 2021 2:53 PM | Permalink | Reply to this

Re: Large Sets 13

Thanks. Fixed.

Posted by: Tom Leinster on July 22, 2021 3:18 PM | Permalink | Reply to this

Re: Large Sets 13

Thanks for writing this series! I found it very interesting and surprisingly accessible.

Posted by: Noah Snyder on July 22, 2021 4:30 PM | Permalink | Reply to this

Re: Large Sets 13

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

Posted by: Tom Leinster on July 22, 2021 4:44 PM | Permalink | Reply to this

Re: Large Sets 13

Even the post abou INaccessible cardinals?

Posted by: Swill stroganoff on July 23, 2021 12:35 AM | Permalink | Reply to this

Re: Large Sets 13

Does “all beths exist” not imply “all alephs exist”? You didn’t draw a line between them on your diagram, but I would have expected there to be an implication there, since W W\aleph_W \le \beth_W.

Posted by: Mike Shulman on July 22, 2021 4:30 PM | Permalink | Reply to this

Re: Large Sets 13

Oh, yes! Thanks. I’ll add that line in.

Posted by: Tom Leinster on July 22, 2021 4:43 PM | Permalink | Reply to this

Re: Large Sets 13

I don’t know if this is an appropriate question. Would it be a problem If I compile a LaTeX copy only for my personal use, merging all the posts? In any case, is a pdf/TeXed copy of this thread going to appear in the near future?


Posted by: V on July 22, 2021 5:18 PM | Permalink | Reply to this

Re: Large Sets 13

Feel absolutely free! The posts are written in Markdown rather than Latex, but you can get the Latex code for individual formulas by double clicking on them. At least, that works in my browser (Firefox).

As for PDF, printing to file from the browser works for me. (I press ctrl-P, then choose “Save to PDF” as the destination.) If you want PDF printouts but can’t get that to work, let me know and I can share mine.

Posted by: Tom Leinster on July 22, 2021 5:41 PM | Permalink | Reply to this

Re: Large Sets 13

Yes, it works (I use FireFox)! Thank you. As for the PDFs, I prefer to export them from LaTeX instead of printing the webpage from browser (I like how tidy LaTex is).

Posted by: V on July 22, 2021 6:41 PM | Permalink | Reply to this

Re: Large Sets 13

Bravo! Congratulations! Many thanks! (Nuff said!)

Posted by: Keith Harbaugh on July 22, 2021 5:55 PM | Permalink | Reply to this

Re: Large Sets 13

Thanks a lot for writing these posts! They were quite understandable and enjoyable.

Posted by: Stéphane Desarzens on July 22, 2021 10:08 PM | Permalink | Reply to this

Re: Large Sets 9.5

I second the above - a very interesting and illuminating series!

Posted by: Jeffrey Ketland on July 22, 2021 11:22 PM | Permalink | Reply to this
Read the post Borel Determinacy Does Not Require Replacement
Weblog: The n-Category Café
Excerpt: Despite what they say.
Tracked: July 24, 2021 1:43 AM

Re: Large Sets 13

No one answered my puzzle, so I’ll give away the answer now: it’s countable measurability.

As you can read at that link, a set is called “countably measurable” (at least by me) if there exists a nontrivial {0,1}\{0, 1\}-valued measure defined on its entire powerset, or equivalently if it admits a nonprincipal ultrafilter closed under countable intersections.

This largeness condition has the property that anything bigger than a large set is large (which is what the puzzle was asking about). For take a countably measurable set XX and a bigger set YY. Choose an injection i:XYi: X \to Y and a suitable measure/ultrafilter on XX. Pushing it forward along ii gives a new one on YY, proving that YY is countably measurable too.

Posted by: Tom Leinster on August 19, 2021 3:55 PM | Permalink | Reply to this
Read the post Large Sets: The Movie
Weblog: The n-Category Café
Excerpt: Video and slides from a talk at UNAM on large cardinals in categorical set theory.
Tracked: November 17, 2021 11:01 PM

Re: Large Sets 13

Oh, I am very much this series can continue towards a powerful cardinals.

At the very least, it should go to Vopěnka’s principle, which has counterparts in the ∞-categories.

The large cardinal theory is not just a painful theory, such as the ω-Woodin cardinals, which are deeply connected to the Axiom of determinacy, which is central to Descriptive set theory.

If possible I would like to know how categorical set theory looks at the inner model / core model. They are elegant, simple, and not as painful as the forcing.

Of course, if I may be allowed to wish, I would very much like to know if there is a counterpart of Reinhardt cardinals within categorical set theory. If I may be allowed to grant a wish, I just want this: Reinhardt cardinals existence is arguably one of the central problems (HOD Hypothesis) of set theory, second only to the continuum hypothesis.

Posted by: Ember Edison on July 27, 2022 2:19 PM | Permalink | Reply to this

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