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July 30, 2021

Diversity and the Mysteries of Counting

Posted by Tom Leinster

Back in 2005 or so, John Baez was giving talks about groupoid cardinality, Euler characteristic, and strange objects with two and a half elements.

two-element group acting on five-element set

I saw a version of this talk at Streetfest in Sydney, called The Mysteries of Counting. It had a big impact on me.

This post makes one simple point: that by thinking along the lines John advocated, we can arrive at the exponential of Shannon entropy — otherwise known as diversity of order 11.

Posted at 11:42 AM UTC | Permalink | Followups (19)

July 28, 2021

Topos Theory and Measurability

Posted by David Corfield

There was an interesting talk that took place at the Topos Institute recently – Topos theory and measurability – by Asgar Jamneshan, bringing category theory to bear on measure theory.

Jamneshan has been working with Terry Tao on this:

  • Asgar Jamneshan, Terence Tao, Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration (arXiv:2010.00681)

The topos aspect is not emphasized in this paper, but it seems to have grown out of a post by Tao – Real analysis relative to a finite measure space – which did.

Posted at 11:44 AM UTC | Permalink | Followups (3)

July 24, 2021

Entropy and Diversity Is Out!

Posted by Tom Leinster

My new book, Entropy and Diversity: The Axiomatic Approach, is in the shops!

If you live in a place where browsing the shelves of an academic bookshop is possible, maybe you’ll find it there. If not, you can order the paperback or hardback from CUP. And you can always find it on the arXiv.

Paperback and hardback with flowers and foliage

I posted here when the book went up on the arXiv. It actually appeared in the shops a couple of months ago, but at the time all the bookshops here were closed by law and my feelings of celebration were dampened.

But today someone asked a question on MathOverflow that prompted me to write some stuff about the book and feel good about it again, so I’m going to share a version of that answer here. It was long for MathOverflow, but it’s shortish for a blog post.

Posted at 8:51 PM UTC | Permalink | Followups (15)

July 23, 2021

Borel Determinacy Does Not Require Replacement

Posted by Tom Leinster

Ask around for an example of ordinary mathematics that uses the axiom scheme of replacement in an essential way, and someone will probably say “the Borel determinacy theorem”. It’s probably the most common answer to this question.

As an informal statement, it’s not exactly wrong: there’s a precise mathematical result behind it. But I’ll argue that it’s misleading. It would be at least as accurate, arguably more so, to say that Borel determinacy does not require replacement.

For the purposes of this post, it doesn’t really matter what the Borel determinacy theorem says. I’ll give a lightning explanation, but you can skip even that.

Thanks to David Roberts for putting me on to this. You can read David’s recent MathOverflow question on this point too.

Posted at 6:32 PM UTC | Permalink | Followups (15)

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.

Posted at 12:41 PM UTC | Permalink | Followups (14)

July 19, 2021

Large Sets 12.5

Posted by Mike Shulman

Previously: Part 12. Next: Part 13

Last time Tom told us about McLarty’s replacement axiom for ETCS, but mentioned that there are several other equivalent axioms in the literature, due to Osius, Lawvere, Cole, and myself. In this addendum I want to discuss and compare those other axioms, and explain why I prefer my own (which is essentially a modification of McLarty’s to a collection axiom rather than a replacement axiom).

Posted at 6:35 PM UTC | Permalink | Followups (25)

July 18, 2021

Large Sets 12

Posted by Tom Leinster

Previously: Part 11. Next: Part 12.5

Today’s topic is replacement. Replacement is not directly about large sets, but it does imply that certain large sets exist.

Even among those who are familiar with and sympathetic to categorical set theory, I think there’s a lingering impression that replacement is somehow borrowed from ZFC. If categorical set theory is supposed to stand on its own two feet, without having to lean on membership-based set theory for conceptual motivation, then perhaps there are those who believe that to supplement ETCS with replacement would be an embarrassing admission of defeat.

I’ll explain why this is a misconception, stating replacement in a way that’s entirely natural from a structural/categorical perspective. The form of replacement I’ll use is due to Colin McLarty, who wrote of it “Our axiom is not a translation from ZF. It is a plain categorical version of Cantor’s idea.”

Posted at 8:04 PM UTC | Permalink | Followups (42)

July 14, 2021

Logical and Sublogical Functors

Posted by John Baez

I’m trying to understand sublogical functors, so I’m looking for examples of sublogical functors between presheaf categories, preferably direct images (if that’s possible).

Just to make this post a bit more interesting, I’ll explain that sentence! This will give beginners a chance to learn something, and experts a chance to catch mistakes in what I’m saying, so that beginners can learn something true.

Posted at 3:03 AM UTC | Permalink | Followups (25)

July 13, 2021

Large Sets 11

Posted by Tom Leinster

Previously: Part 10. Next: Part 12

Measurability is the largest of the “large set” conditions I’m going to talk about in this series. Today I’ll explain how measurability relates to inaccessibility, say a tiny bit about how measurability can arise in analysis problems, and say somewhat more about measurability and codensity monads.

Posted at 5:34 PM UTC | Permalink | Followups (19)

July 10, 2021

Large Sets 10

Posted by Tom Leinster

Previously: Part 9.5. Next: Part 11

The early decades of the 20th century saw the development not only of axiomatic set theory, but also of Lebesgue’s theory of integration and measure. At some point, the two theories met and gave birth to the notion of measurability for sets. Measurability is maybe the most appealing of the “large set” conditions: it’s important set-theoretically, natural categorically, and — true to its origins — continues to arise occasionally in actual analysis.

Posted at 9:47 PM UTC | Permalink | Followups (6)

July 8, 2021

Large Sets 9.5

Posted by Mike Shulman

Previously: Part 9. Next: Part 10

In the last comment thread, Tom invited me to write a post about some of the sizes of sets in between inaccessible sets and measurable sets. I’m not sure he was serious, but I’m going to take him up on it anyway. (-:

There are a lot of sizes of sets in between inaccessibles and measurables, but in this post I’ll just talk about “higher inaccessible” sets and Mahlo sets. I think these are worth thinking a bit about, especially as a followup to Tom’s very nice description of various kinds of large sets that are smaller than inaccessibles, because they can be thought of roughly as continuing the project of “making things that can’t be reached from below”. Measurable sets and their ilk feel to me like less of a straightforward continuation of that project, bringing in somewhat more exotic definitions that turn out to make them very large.

In addition, I hope to give a very fragmentary idea of how must vastly bigger than an inaccessible set a measurable set must be, by exploring just a bit of the terrain in between.

Posted at 4:02 AM UTC | Permalink | Followups (30)

July 6, 2021

Large Sets 9

Posted by Tom Leinster

Previously: Part 8. Next: Part 9.5

Today I’ll talk about inaccessibility. A set is said to be “inaccessible” if it cannot be reached or accessed from below using certain operations. We’ve seen this rough idea before — but which operations are the ones in play here, and what makes them especially interesting?

Posted at 10:55 AM UTC | Permalink | Followups (6)

July 2, 2021

Large Sets 8

Posted by Tom Leinster

Previously: Part 7. Next: Part 9

If you’ve been wanting to follow this series but haven’t had time to keep up, now’s a good moment to hop back on board — I won’t assume much of what’s gone before.

Back in the mists of time, when I took a first undergraduate course on axiomatic set theory, I was exhilarated by the extraordinary world of infinite sets I saw opening up before me. In that world, addition is the same as multiplication! Which is the same as maximum! That is,

X+YX×Ymax(X,Y) X + Y \cong X \times Y \cong max(X, Y)

for all infinite XX and YY. It seemed unthinkably exotic.

I then heard this part of cardinal arithmetic called “trivial” for exactly the reasons just stated. Although that description is technically correct, it poured a bucket of cold water over my enthusiasm in a way that only mathematicians can.

So with apologies to my past self, I give you the informal title of this post: the nontrivial part of cardinal arithmetic.

Posted at 1:26 PM UTC | Permalink | Followups (17)

June 29, 2021

Large Sets 7

Posted by Tom Leinster

Previously: Part 6. Next: Part 8

Given a well-ordered set WW, there are at least two ways of manufacturing a plain, unadorned set. You can, of course, take the underlying set U(W)U(W). But you can also take the beth W\beth_W. How do they compare in size?

Let’s look at some of the first few cases, recalling that when nn is a natural number, n\beth_n means W\beth_W for the unique well-ordered set WW with nn elements.

  • 0\beth_0 is \mathbb{N}, the smallest infinite set, whereas 00 is the empty set.

  • 1\beth_1 is the uncountable set 2 2^\mathbb{N}, whereas 11 is the well-ordered set with only one element.

  • 4=2 2 2 2 \beth_4 = 2^{2^{2^{2^{\mathbb{N}}}}} is probably bigger than any specific set used by 95% of mathematicians in a lifetime, whereas 44 has, well, just four elements.

So comparing W\beth_W with U(W)U(W) seems like racing an intercontinental ballistic missile against a snail — or more traditionally, a hare against a tortoise.

Unlike in the fable, our tortoise never overtakes the hare. But it’s conceivable that it does keep catching up, only to fall behind again an instant later. Moments when the tortoise catches the hare are called “beth fixed points”, and they’re our topic for today.

Posted at 3:47 PM UTC | Permalink | Followups (16)

June 24, 2021

Large Sets 6

Posted by Tom Leinster

Previously: Part 5. Next: Part 7

The plan for this series is to talk about ever larger sets and ever stronger axioms. So far we’ve looked at weak limits, strong limits, and alephs. Today we’ll look at beths.

The beths are the sets you get if you start with \mathbb{N} and repeatedly take power sets. They are

0=, 1=2 , 2=2 2 , \beth_0 = \mathbb{N}, \ \beth_1 = 2^\mathbb{N}, \ \beth_2 = 2^{2^{\mathbb{N}}}, \ \ldots

“and so on”, with one set W\beth_W for each well-ordered set WW. The symbol \beth is beth, the second letter of the Hebrew alphabet, after aleph. And like the alephs, the beths aren’t all guaranteed to exist.

Posted at 3:45 PM UTC | Permalink | Followups (5)