Anthropomorphizing the Russell paradox
Anthropormorphization in mathematics—an excerpt from my podcast with Lex Fridman, a sweeping conversation on infinity, philosophy, and mathematics.
I sat down recently for a sweeping conversation with Lex Fridman on infinity, paradoxes, philosophy, mathematics, and more.
At one point, we talked about an instance of my advice to young mathematicians on the value of anthropomorphization as a means to achieve mathematical insight, applied in the case of Cantor’s argument on uncountable infinities and the Russell paradox. Please enjoy the excerpt—the transcript appears below.
Lex Fridman(00:49:23) You mentioned to me offline we were talking about Russell’s paradox and that there’s another kind of anthropomorphizable proof of uncountability. I was wondering if you can lay that out.
Joel David Hamkins(00:49:41) Oh yeah, sure. Absolutely.
Lex Fridman(00:49:42) Both Russell’s paradox and the proof.
Joel David Hamkins(00:49:44) Right. So we talked about Cantor’s proof that the real numbers, the set of real numbers is an uncountable infinity, it’s a strictly larger infinity than the natural numbers. But Cantor actually proved a much more general fact, namely that for any set whatsoever, the power set of that set is a strictly larger set. So the power set is the set containing all the subsets of the original set. So if you have a set and you look at the collection of all of its subsets, then Cantor proved that this is a bigger set. They’re not equinumerous. Of course, there’s always at least as many subsets as elements because for any element, you can make the singleton subset that has only that guy as a member, right? So there’s always at least as many subsets as elements.
Joel David Hamkins(00:50:36) But the question is whether it’s strictly more or not. And so Cantor reasoned like this. It’s very simple. It’s a kind of distilling the abstract diagonalization idea without being encumbered by the complexity of the real numbers. So we have a set X and we’re looking at all of its subsets. That’s the power set of X. Suppose that X and the power set of X have the same size, suppose towards contradiction, they have the same size. So that means we can associate to every individual of X a subset. And so now let me define a new set. I mean, another set, I’m going to define it. Let’s call it D. And D is the subset of X that contains all the individuals that are not in their set.
Joel David Hamkins(00:51:28) Every individual was associated with a subset of X, and I’m looking at the individuals that are not in their set. Maybe nobody’s like that. Maybe there’s no element of X that’s like that, or maybe they’re all like that, or maybe some of them are and some of them aren’t. It doesn’t really matter for the argument. I defined a subset D consisting of the individuals that are not in the set that’s attached to them, but that’s a perfectly good subset. And so because of the equinumerosity, it would have to be attached to a particular individual, you know? And- Let’s call that person, it should be a name starting with D, so Diana.
Joel David Hamkins(00:52:10) And now we ask, is Diana an element of D or not? But if Diana is an element of D, then she is in her set. So she shouldn’t be because the set D was the set of individuals that are not in their set. So if Diana is in D, then she shouldn’t be. But if she isn’t in D, then she wouldn’t be in her set. And so she should be in D. That’s a contradiction. So therefore, the number of subsets is always greater than the number of elements for any set. And the anthropomorphizing idea is the following. I’d like to talk about it this way. For any collection of people, you can form more committees from them than there are people, even if you have infinitely many people.
Joel David Hamkins(00:53:03) Suppose you have an infinite set of people, and what’s a committee? Well, a committee is just a list of who’s on the committee basically, the members of the committee. So there’s all the two-person committees and there’s all the one-person committees and there’s the universal, the worst committee, the one that everyone is on. Okay. The best committee is the empty committee. With no members and never meets and so on. Or is the empty committee meeting all the time? I’m not sure.
Lex Fridman(00:53:29) Yeah. That’s… wow, that’s a profound question. And does a committee with just one member meet also?
Joel David Hamkins(00:53:35) Yeah. Maybe it’s always in session. I don’t know. So the claim is that there are more committees than people. Okay. Suppose not. Well, then we could make an association between the people and the committees. So we would have a kind of… every committee could be named after a person in a one-to-one way. And I’m not saying that the person is on the committee that’s named after them or not on it, whatever. Maybe sometimes that happens, sometimes it doesn’t. I don’t know. It doesn’t matter. But let’s form what I call committee D, which consists of all the people that are not on the committee that’s named after them.
Joel David Hamkins(00:54:16) Okay. Maybe that’s everyone, maybe it’s no one, maybe it’s half the people. It doesn’t matter. That’s a committee, it’s a set of people. And so it has to be named after someone. Let’s call that person Daniella. So now we ask, is Daniella on the committee that’s named after her? Well, if she is, then she shouldn’t be because it was the committee of people who aren’t on their own committee. And if she isn’t, then she should be. So again, it’s a contradiction. So when I was teaching at Oxford, one of my students came up with the following different anthropomorphization of Cantor’s argument. Let’s consider all possible fruit salads. We have a given collection of fruits.
Joel David Hamkins(00:55:07) You know, apples and oranges and grapes, whatever. And a fruit salad consists of some collection of those fruits. So there’s the banana, pear, grape salad and so on. There’s a lot of different kinds of salad. Every set of fruits makes a salad, a fruit salad. Okay… And we want to prove that for any collection of fruits, even if there are infinitely many different kinds of fruit, for any collection of fruits, there are more possible fruit salads than there are fruits. So if not, then you can put a one-to-one correspondence between the fruits and the fruit salads, so you could name every fruit salad after a fruit. That fruit might not be in that salad, it doesn’t matter. We’re just… it’s a naming, a one-to-one correspondence.
Joel David Hamkins(00:55:53) And then, of course, we form the diagonal salad, which consists… Of all the fruits that are not in the salad that’s named after them. And that’s a perfectly good salad. It might be a kind of diet salad, if it was the empty salad, or it might be the universal salad…
Joel David Hamkins(00:56:12) which had all fruits in it, if all the fruits were in it. Or it might have just some and not all. So that diagonal salad would have to be named after some fruit. So let’s suppose it’s named after durian, meaning that it was associated with durian in the one-to-one correspondence. And then we ask, well, is durian in the salad that it’s named after? And if it is, then it shouldn’t be. And if it isn’t, then it should be. And so it’s, again, the same contradiction. So all of those arguments are just the same as Cantor’s proof that the power set of any set is bigger than the set.
Joel David Hamkins(00:56:48) And this is exactly the same logic that comes up in Russell’s paradox, because Russell is arguing that the class of all sets can’t be a set because if it were, then we could form the set of all sets that are not elements of themselves. So basically, what Russell is proving is that there are more collections of sets than elements. Because we can form the diagonal class, you know, the class of all sets that are not elements of themselves. If that were a set, then it would be an element of itself if and only if it was not an element of itself. It’s exactly the same logic in all four of those arguments. So there can’t be a class of all sets, because if there were, then there would have to be a class of all sets that aren’t elements of themselves.
Joel David Hamkins(00:57:40) But that set would be an element of itself if and only if it’s not an element of itself, which is a contradiction. So this is the essence of the Russell paradox. I don’t call it the Russell paradox. Actually, when I teach it, I call it Russell’s theorem. There’s no universal set. And it’s not really confusing anymore. At the time, it was very confusing, but now we’ve absorbed this nature of set theory into our fundamental understanding of how sets are, and it’s not confusing anymore. I mean, the history is fascinating though, about the Russell paradox, because before that time, Frege was working on his monumental work undertaking, implementing the philosophy of logicism, which is the attempt to reduce all of mathematics to logic.
Joel David Hamkins(00:58:30) So Frege wanted to give an account of all of mathematics in terms of logical notions, and he was writing this monumental work and had formulated his basic principles. And those principles happened to imply that for any property whatsoever, you could form the set of objects with that property. This is known as the general comprehension principle. And he was appealing to the principles that support that axiom throughout his work. I mean, it was really… It wasn’t just an incidental thing, he was really using this principle.
Joel David Hamkins(00:59:11) And Russell wrote him a letter when he observed the work in progress, that there was this problem, because if you accept the principle that for any property whatsoever you can make a set of objects with that property, then you could form the set of all sets that are not members of themselves. That’s just an instance of the general comprehension principle. And… But the set of all sets that aren’t elements of themselves can’t be a set, because if it were, then it would be an element of itself if and only if it’s not a member of itself, and that’s a contradiction. And so Russell wrote this letter to Frege, and it was just at the moment when Frege was finishing his work. It was already at the publishers and, you know, in press basically. But it’s completely devastating.
Joel David Hamkins(00:59:58) I mean, it must have been such a horrible situation for Frege to be placed in, because he’s finished this monumental work, you know, years of his life dedicated to this, and Russell finds this basically one-line proof of a contradiction in the fundamental principles of the thesis that completely destroys the whole system. And Frege had put in the appendix of his work a response to Russell’s letter in which he explained what happened, and he wrote very gracefully, “Hardly anything more unwelcome can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This is the position into which I was put by a letter from Mr.
Joel David Hamkins(01:00:46) Bertrand Russell as the printing of this volume was nearing completion.” And then he goes on to explain the matter, it concerns his basic law five and so on.
Lex Fridman(01:00:54) It’s heartbreaking. I mean, there’s nothing more traumatic to a person who dreams of constructing mathematics all from logic, to get a very clean, simple contradiction. I mean, that’s just…
Joel David Hamkins(01:01:08) You devote your life to… This work, and then it’s shown to be contradictory, and that must have been heartbreaking.
Lex Fridman(01:01:16) What do you think about the Frege project, the philosophy of logic, the dream of the power of logic… To construct a mathematical universe?
Joel David Hamkins(01:01:24) So, of course, the project of logicism did not die with Frege, and it was continued, and, you know, there’s a whole movement, the neologicists and so on, in contemporary times even. But my view of the matter is that really, we should view the main goals of logicism are basically completely fulfilled in the rise of set-theoretic foundationalism. I mean, when you view ZFC as the foundation of mathematics, and in my view, the principles of ZFC are fundamentally logical in character, including the axiom of choice, as I mentioned, as a principle of logic. This is a highly disputed point of view, though, because a lot of people take even the axiom of infinity as mathematical, inherently mathematical and not logical and so on.
Joel David Hamkins(01:02:14) But I think if you adopt the view that the principles of ZFC have to do with the principles of abstract, you know, set formation, which is fundamentally logical in character, then it’s complete success for logicism. So the fact that set theory is able to serve as a foundation means that mathematics can be founded on logic.
See the full transcript and watch the full video episode for more. I shall periodically be posting more excerpts like this one here on Infinitely More—find them in the lex-fridman tag.