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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.
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 .
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.
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.
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.
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.
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.
- Part 1: introduction
- Part 2: limits
- Part 3: well-ordered sets
- Part 4: the index of a set
- Part 5: alephs
- Part 6: beths
- Part 7: beth fixed points
- Part 8: cardinal arithmetic, including regularity and cofinality
- Part 9: inaccessibility
- Part 9.5 (by Mike Shulman): hyper-inaccessibility and Mahloness
- Part 10: measurability
- Part 11: measurability, continued
- Part 12: replacement
- Part 12.5 (by Mike Shulman): comparison of replacement axioms
- Part 13: summary.
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).
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.”
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.
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.
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.
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.
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?
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,
for all infinite and . 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.
June 29, 2021
Large Sets 7
Posted by Tom Leinster
Previously: Part 6. Next: Part 8
Given a well-ordered set , there are at least two ways of manufacturing a plain, unadorned set. You can, of course, take the underlying set . But you can also take the beth . How do they compare in size?
Let’s look at some of the first few cases, recalling that when is a natural number, means for the unique well-ordered set with elements.
is , the smallest infinite set, whereas is the empty set.
is the uncountable set , whereas is the well-ordered set with only one element.
is probably bigger than any specific set used by 95% of mathematicians in a lifetime, whereas has, well, just four elements.
So comparing with 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.
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 and repeatedly take power sets. They are
“and so on”, with one set for each well-ordered set . The symbol is beth, the second letter of the Hebrew alphabet, after aleph. And like the alephs, the beths aren’t all guaranteed to exist.