My Favorite Theorem

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn lamb, a freelance math and science writer in Salt Lake City, Utah, which is very chilly, and I'm trying to warm up from my bike ride just now that I probably should have worn better gloves on. The downhill!

KK: Yeah, yeah, that sweater you're wearing those should help this. Yes.

EL: Yes, it is. I'm showing all of our listeners that this is a wonderfully rainbow chunky sweater that I recently inherited from my grandmother. Probably purchased in 1973 or something.

KK: Yeah. Well, so today's actually, it's November 11. It’s Veterans Day and the University of Florida is closed and so I've done wonderful things, like I went to brunch with my wife this morning. We sat outside. It's not cold here, of course. We sat outside, and then I fixed her, the light in her closet, it was one of those pull cord deals, you know, the core just broke. I mean, inside the lamp, so I had to actually go to Ace Hardware, which I highly recommend. Your local Ace is much better than the big box places and and get the stuff to fix it. So that's been my day.

EL: Thanks to our sponsors, Ace Hardware. No. They care so much about math podcasts, I’m sure.

KK: I’m sure. All right. Well, today we are pleased to welcome Kameryn Williams. Kameryn, why don’t you introduce yourself?

Kameryn Williams: Hi, I’m Kameryn Williams. I'm a mathematician at Sam Houston State University about an hour north of Houston, Texas. It's good to be on.

EL: Yeah, and I bet it's nice and warm there where you are.

KW: It’s in the 70s today, you might think that's warm, being in Salt Lake.

EL: Yeah, I went to grad school in Houston. So I do — November, February, March. Great months there.

KW: Oh, yeah, definitely. Well, like I have family in Idaho, and they've been complaining they got snow recently. And I'm like, come on, it’s not that bad. Quit complaining.

EL: Did you grow up in a warm place? Or?

KW: I spent some time in California growing up but moved around a fair bet.

EL: Okay. Yeah.

KK: All right. Well, so what sort of math are you interested in?

KW: I’m a logician, mostly specializing in set theory.

KK: Oh, okay. Well, that’s, that’s —

EL: I guess that’s okay.

KW: You’ve had a few logicians on, I’ve seen.

KK: We have.

EL: We just had your advisor on a couple of months ago.

KW: That’s true, Joel [David Hamkins] was on. So he took my favorite theorem, so I had to come up with a new one.

EL: So rude.

KK: Oh, well, okay. Well, that's the perfect segue. So what is it? What is your what do you want to talk about today?

KW: Okay. Yeah, so I guess maybe I should hedge? It's hard for me to pick favorites. It's like your children. They're all my favorite.

KK: Sure. Sure.

KW: But definitely among my favorites, one of them is Gödel’s condensation lemma.

KK: Okay, so I don't know this one. I mean, everybody knows his famous incompleteness theorems, right, but —

KW: Yeah, so this comes, I don't know, what, about 10 years after his incompleteness theorems. Okay, so maybe I should step back a bit and not talk about Gödel for a second and kind of set the stage for the theorem.

KK: Okay, that would be great.

EL: That would be fantastic.

KW: By itself, you kind of need to know why it's there to understand why anyone would love it. So set theory, like the study of well-foundedness, we kind of care about these transfinite iterative constructions. So here's a good example: Let’s generate all the sets. You can start with whatever non-sets you have — numbers, whatever — and then you just look at sets of those, sets of sets of those, sets of sets of sets of those, and so on transfinitely. So you just iterate the power set operator, along the ordinals transfinitely out to infinity, you get all the sets that way.

Okay, so it looks like if we want to do this, we need all these sorts really large, weird infinite objects going on. But 1922, Thoralf Skolem made an interesting theorem now known as Skolem’s paradox — whether you think it's a paradox is on you — and he noticed that you could just have a single small countable collection of things that had all the properties what it looks like you needed all these really large sets for. So there's this small little countable thing, it’s got what it thinks are all the real numbers. You know, externally you can see it's countable, but internally, it doesn't know that. It thinks there's uncountably many real numbers. You can do everything with that.

EL: Whoa. Sorry, so you can tell I've not done much with this kind of stuff here, so you’re blowing my mind here.

KK: This is good because that’s kind of how it was when I first learned this. Like, if you would have brought me on as an undergrad, I would have said downward Löwenheim-Skolem, that’s my favorite theorem. Given any structure, you can find a countable elementary substructure, a countable thing that has all the same sort of intrinsic properties.

EL: Okay.

KK: Okay, so from the outside the reals look countable, but inside they don’t?

KW: Yeah, well, so what does it mean that the reals are—

EL: That seems way too powerful.

KW: What does it mean that something’s countable? It means there exists a bijection with the natural numbers. Well, if you don't have all the objects, you just have a small piece, you might be missing that bijection.

KK: Okay.

KW: So you might think it's uncountable, but it's just because your model was too small to see all the bijections.

KK: Okay.

KW: Okay, so Skolem, he called it a paradox. He thought this, you know, revealed all the set theory people were doing was nonsense. Like come on, you can’t even nail down what it means for something to be uncountable. And I mean, the other people kind of had your reaction, you know, it’s blowing my mind. That’s kind of where I fell. But it was really not treated as anything more than a curiosity. So maybe you thought it was a paradox, maybe you thought it was curiosity, but no one really had any use for it beyond that.

And then, late 1930s, about 15 to 20 years after Skolem did this, Gödel was interested, you know, he was fresh off the incompleteness theorems. Everyone loves those. He wants to go onto something a bit more difficult. And so he's concerned about Hilbert’s first problem. Can you prove Cantor's continuum hypothesis? Can you prove that there's no cardinality intermediate between the natural numbers and the real numbers?

KK: Right.

EL: Right.

KW: Okay. Skolem says, you know, maybe it's hard to say what that means, but let's set that aside. Okay, so how is he going to do this is he wants to do something like this construction I mentioned, iterate the power set, but you want to have a bit more control. The problem is if you take the power of sort of the natural numbers, that's going to contain basically every countable object, and how do you get any sort of handle and what this says? So Gödel’s idea was to restrict this to as small as possible to just the intrinsic things. So rather than taking all the sets of natural numbers, you just want to take the ones that you can define just by quantifying over natural numbers. So you'd have, like, the set of prime numbers, because you can say what it means for something to be prime, you can have the set of even numbers, you can say that, but you wouldn't have a weird thing, for example, like these, you know, weird countable models of set theory, you can't really define that just by quantifying over natural numbers, okay, or isomorphic copies of them, however you want to say it.

KK: Okay.

KW: So he iterates this process where you just take the definable sets. And then if you iterate that transfinitely instead of the full power set operator, you get some other hierarchy, some other transfinite hierarchy of construction. So this is called the L hierarchy. I looked this up because I didn't know: L stands for law. Not entirely sure why he chose law.

EL: Like, rules, that kind of law?

KW: Yeah, I'm not entirely sure what he thought with it, but that's what it was. Okay. So just like with this, like, iterating the power set hierarchy you can do like a sort of Skolem collapse there. You can do the same thing with Gödel’s L hierarchy. So you can build it up to a certain stage. And then you can say, okay, I built it up to some uncountable stage. I'm going to take just a countable piece that looks like this. It's got all the same elementary, or to use a bit of jargon, first-order properties.

KK: Okay.

KW: Okay, so Gödel’s condensation lemma just says that if you do this Skolem collapse process, take this elementary sub-model, it's just going to be isomorphic to an initial segment of the construction. So maybe you built up to stage alpha, you take this countable piece, it's like building up to stage beta, some stage beta below of alpha.

KK: Okay.

KW: Okay. And looking at your faces, it's not clear why this is interesting.

EL: Right. Well, I do have a question that — sometimes it's really fun to be very naive and ask ridiculous questions. So like, okay, if we're trying to do this process for the real numbers, say, and find this countable thing underneath it, like, is there a canonical way to say, okay, these are the real numbers that are in this countable piece of it. Like, can I say is pi in here is, is 1/4 in here? Could I ask you that? And you could say yes or no.

KW: I can say yes to both pi and 1/4. Other numbers might be harder to say. But those are easy. Yeah.

EL: All right.

KW: This is the point, is you have a very firm control over what's going on. This is kind of the opposite of taking the power set construction where, what is the power set of the natural numbers? That's kind of the vague question that the continuum hypothesis is trying to get at. This is we're going to get at it very firmly. But this is interesting because Skolem’s paradox gives us this condensation lemma when you apply it in Gödel’s constructible universe. So this paradox becomes a theorem. And this theorem then is kind of the key lemma to prove the continuum hypothesis for L. So you want to now prove that you can't have too many reals. Well, any real you have, it shows up at some countable stage. So if every real shows up at a countable stage, there's aleph-one many countable stages. Aleph-one is the smallest uncountable cardinal. That means there's aleph-one many reals, the smallest uncountable cardinal, nothing in between.

EL: I feel like you're pulling one over on us.

KK: I don't think that's a proof of the continuum hypothesis because, you know —

KW: It’s a proof of the continuum hypothesis in Gödel’s model he built.

KK: Okay.

KW: So this is how he showed, then, that it's, you can't disprove it from this.

EL: Right. It's not a contradiction.

KW: Yeah. So it's consistently true because it's true here. Now, this isn't necessarily true in the full universe of sets. You might miss out on a lot of things in Gödel’s construction, and then later Paul Cohen showed, okay, you can't prove it for the full universe of sets.

KK: Right.

KW: But yes, this is just one step, this thing that was a paradox became a key lemma in order to kind of do half of Hilbert first problem. And I think it's kind of interesting, something that’s first is introduced, maybe it's a paradox, maybe it's a cute little curiosity, and then Gödel, you know, 15, 20 years later or so, realizes, oh, no, this is actually — a version of this is a really important step in this proof.

KK: Hm. Okay, this is completely foreign to me.

EL: Yeah. Well, I'm a little confused on the timeline here. Had Gödel already done the incompleteness theorems, but then this is something else that he did later?

KW: This is after. Yeah, yes. The stuff with L, this is about 1940. Incompleteness theorems are about 1930. So it's about 10 years later.

EL: Okay. But he kind of figured out that this work fits in with the incompleteness theorems.

KW: Well, I mean, the incompleteness theorems are one of the first forays into a larger program. Like, okay, you know, they established, you can't write down a computable list of axioms that decides all the things you want. Well, can you then find natural questions that you can decide? Well, here's one that was a candidate: the continuum hypothesis. So it's easy to say, okay, here's these weird diagonalization tricks to get something. How do you show that a “natural statement” is undecidable? That takes more work. It’s a more difficult problem.

KK: Okay.

EL: Yeah. And so we — I emailed you to be on the show, because I saw a tweet you had done about how maybe you felt it was a little unfair that Gödel is so known for the incompleteness theorems, whereas you feel like this work is actually more impressive, and people should be more, I guess, should like it more than the incompleteness theorems.

KW: Ah, that was maybe a bit of a hot take. But I think it's it's uncontroversially mathematically more impressive. The proof of the incompleteness theorems is basically he noticed the same sort of similar sort of weird thing people had kind of dismissed, and he realized this could be important, this diagonalization trick. If you can somehow code proofs and formulas into number theory, they can refer to themselves and you get this self-reference. So, you know, cute little curiosity, and he takes this cute little curiosity and he turns into a theorem. So he did the same thing here, but it was a much more difficult theorem, much harder to do. And not to pooh-pooh the incompleteness theorems, but if you take a senior undergraduate-level logic class, you'll prove the incompleteness theorems by the end. They’re not that difficult of results.

KK: Well, that's probably why they get more love, right?

KW: Yeah. I mean, that's a big part of it. And so that's maybe where a bit of a hot take came from.

KK: Right, right. Yeah, this is all very Kafkaesque to me, right? So, you know, Kafka has these — he uses infinity a lot in his in his writing, you know. It reminds me of — so the L standing for law made me think of Kafka’s story “Before the Law,” which is part of The Trial.

KW: I’m not familiar with that one.

KK: It’s part of The Trial, but it's often pulled out as a separate story. So the deal is that the man goes to see the law, and there's a gatekeeper, and he can't get through because the guy won't let him through. But he says even if you do, there's another door inside. It’s sort of like these infinite gatekeepers that you can never, so you can never get to the law. And then the man who came to see the law, eventually he just stays because he needs to get in there, but he can never get in. And he keeps asking the guard, and the guard says, eh, maybe tomorrow. And then finally he just dies. You know, it's very, it is a very Kafka sort of story. But in a way, sort of these hierarchies are embedded in all of that. Okay. So I mean—

KW: I mean, what he should have done is he should have taken an elementary sub-model.

EL: Yeah.

KW: Don't wait till the very end. Just get it down to a smaller piece.

EL: Exactly. We're about to publish “Mathematicians Rewrite Kafka.” Mathematicians rewrite all 20th century literature, solve any problems anyone ran into, just take care of it.

KK: So you've loved this theorem for how long? Is this a love at first sight kind of thing?

KW: Ah, I mean, I'll be honest, it took me a good amount of time to really appreciate it. And you kind of see the statement in a textbook, and it's this abstract technical thing, and it's really, I mean, okay, if I'm being honest, Kanamori, Aki Kanamori, has some really nice papers that go through the history of this, and reading those made me appreciate a lot more, seeing where it came fromm some of the intellectual development, rather than just after the fact, here is this lemma in the middle of some technical exposition.

EL: Yeah, well, and often, when you're seeing theorems in a textbook, it's like, they don't have a flag next to the one that like you should really pay attention to or love. I remember, when I first took, like, my first algebraic topology class, there was — this wasn't a theorem, it was a definition — called the fundamental group. And it wasn't clear to me for a while that like, this was a particularly important object. And then a couple of weeks later, I was like, man, two weeks later, we're still talking about this fundamental group? And sure, it has the name fundamental, so that part is on me that I didn't really notice that. But yeah, sometimes when you're reading, you know, the first time you encounter this, you're just deluged with all this information, and you’re making your own hierarchy of which things you know, and how the information fits together, and which things are really central, is difficult.

KW: Well, I mean, if it was easy, they wouldn't pay us to do it.

EL: Right.

KK: That’s true. So the other thing we do on this podcast is we ask our guests to pair their theorems with something. So what pairs well with Gödel’s condensation lemma.

KW: Ah, I would say probably a nice brie with some green apples and some honey.

EL: Okay, that does sound delightful.

KW: I mean, I don't think it has particularly anything to do with the theorem, but I just think brie is nice.

KK: You just want to eat that anytime, right? Oh, yeah. I'm there with you.

EL: I mean, yeah, that sounds good. While you're reading Kafka also?

KW: Maybe a nice wine too. Okay. All right. Yeah.

EL: Well, that does sound just lovely.

KW: So we always like to give our guests a chance to, to plug something. Where can we find you on the worldwide intertubes?

KW: My website is just my name, KamerynJW.net. Okay, you can spell it because it will be linked in the episode description, presumably.

KK: We’ll link straight there.

Yeah, the thing I would like to plug as I'm on the job market. Please hire me. I’ll just be honest. That's what I would like to plug.

KK: Okay. Great.

EL: Excellent.

KK: Everyone needs work.

EL: Yeah. Any hiring committees that ask us for references, we'll send in this episode.

KW: Yeah. Just I would like a tenure track job.



KK: Okay. How long have you been at Sam Houston? KW: This is my first year here. Previously, I was at University of Hawaii for a postdoc.

KK: Okay. That's quite the switch.

KW: A little bit, but you know, they offered me a position there. Like, I’m not going to say no to a postdoc in Hawaii.

KK: No, no, no.

KW: If I don't like it, I just don't stay. It’s a postdoc.

KK: And Sam Houston is in Huntsville, correct?

KW: Yeah.

KK: Okay. So yeah, different vibe.

KW: A little bit different, yeah.

EL: Slightly. Yeah. Do you live in Huntsville or in Houston?

KW: I live in town. It's a bit too far to commute every day.

EL: Yeah. I know, actually, a friend of mine, who is one of your colleagues I know lives in Houston because her spouse has a job in Houston. And so they drive opposite directions in the morning, but that sounds hard.

KK: Yeah, well and Houston is so big. You could still live in Houston and have like a two hour commute, right?

KW: Oh, yeah. Right. It's an hour to Houston, but that's just the outskirts of Houston and depending on where you want to go, it might be two or more.

EL: Great city, though. I miss living in Houston, even though people like to trash it for being such urban sprawl, and muggy and humid and full of mosquitoes and everything, but I liked it.

KW: My problem is I did grad school in New York. So I see any other large city and I'm like, Here are the ways this is not like New York. I don't like that.

KK: Well, New York's dense, right?

KW: Yeah, exactly.

KK: I once knew a mathematician who got a job in LA and and he complained that it wasn't big enough for him. He wanted to live in Tokyo or someplace like that. I just thought, well, okay, more power to you. Here. I am in my little, you know, 100,000 person town in Florida.

KW: I mean, that's larger than where I am.

KK: Sure. All right. Well, this has been this has been great fun, Kameryn. I learned something, I think.

EL: Yeah.

KK: I have to turn it over. Yeah.

EL: Yeah. One thing I love about writing the transcripts for the episodes is I get to, you know, experience it again. And then I often make a lot of connections. Like, oh, it would have been really smart of me to ask a question about this. But in the moment, I didn't think of it. So you know, everyone could listen to it again and get something else out of it.

KK: Right.

EL: Thanks for joining us.

KW: Yeah. Thank you for having me.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking with Kameryn Williams from Sam Houston State University about Gödel's condensation lemma. Here are some links you might find interesting after you listen to the episode.
Their website and Twitter account
The Skolem paradox in the Stanford Encyclopedia of Philosophy
Gödel's incompleteness theorems in the Stanford Encyclopedia of Philosophy
Akihiro Kanamori's paper about Gödel and set theory
A shorter and more accessible paper by Kanamori and Juliet Floyd on the same topic
Before the Law by Franz Kafka