Doodling the Axioms of Set Theory

Our son is almost three, and he’s starting to really like to draw. He’s also getting to the age where, if he doesn’t have something to do, he’ll tear our apartment to shreds, so most Sunday afternoons we dump a lunchbox of crayons and whatever paper we have on the floor. It’s arts and crafts time, Yosef!

Now, don’t get me wrong, I like watching my kid draw as much as the next parent. (“It’s a fish? No, a dog? Oh, wow, that’s terrific.”) But, what can I say, I get a bit bored.

More to the point, drawing with crayons is so much fun. Arts and crafts time is great. So I draw along with him. And very often I find myself trying to doodle some math.

Lately I’ve been studying a book called Classic Set Theory. It’s been really working for me. It has great exercises, clear organization, oodles of historical context. It’s subtitled “For Guided Independent Study,” and it’s making me wonder why every math textbook isn’t for guided independent study too.

Set theory and logic was my way back into math. I was a philosophy major and had all sorts of worries about whether I could handle a college math class. Then I took a philosophy class about Frege, Russell and Wittgenstein, figures who stand at the creation of mathematical logic and set theory.

In that class, I was finally starting to understand how I had to study this stuff — line by line, ask myself lots of questions, don’t leave anything fuzzy. (Which is more of less how I know to study Talmud.) And I was realizing that if I put in this sort of effort, and if I was honest with myself about what I got and what I didn’t, I could understand some (if I do say so) ridiculously abstract stuff at at least a basic level.

“At some level,” because there was still a lot of stuff that I didn’t get. Since college, set theory and logic has been some of the math that I’ve read the most on. But I still haven’t felt like I really understood set theory, as I hadn’t been able to make much headway in any set theory text.

This is part of why Classic Set Theory is so much fun for me to read. Finally, I feel like I’m starting to get this stuff. Finally!

Here’s the question I found myself asking while drawing with my son today: what could the axioms of set theory* look like if you drew them?

The axioms of Zermelo-Fraenkel set theory with the Axiom of Choice, I mean. I’d love to understand some of the other set theories better. During that class I took in college we studied type theory as Russell’s attempt to patch up the contradiction he discovered in Frege’s system, but I think I only ever had a surface-level understanding of what this meant.

There are nine axioms in all. I know that explanations work better when the objects to be explained are doing stuff, so I tried to make the axioms as active as possible in the drawings. In practice, this means I interpreted the axioms as little machines, rather than as rules or laws. (Axioms have always been presented to me as rules, and until the last couple of days that’s always how I’d always thought of them.)

The toughest one to visually represent was Replacement. The ones I’m most worried about having misrepresented are the Axiom of Infinity and of Foundation. Honestly, all of them are probably flawed in some way. There might be mathematical errors or misinterpretations — as long as you’re nice about it, would you point those out to me?

But I’m not sharing these as resources or to make some point about teaching. I’m sharing these out of celebration, and a bit of relief, that I might be finally figuring out how to learn some math on my own.

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Extensionality, Empty Set, and the Axiom of Pairs
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Axiom of Separation
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Power Set Axiom
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Union Axiom
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Axiom of Replacement
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Axiom of Infinity
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Axiom of Foundation
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Axiom of Choice
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9 thoughts on “Doodling the Axioms of Set Theory

    1. Let me see if some additional words can explain what I was going for. I was trying to send the picture for the “axiom of infinity” backwards. Meaning, say that you have some infinite set — there is no infinite descent. That set always HAS to have an element that isn’t a member of anything else in the set. I think of that foundation, rock-bottom element as the empty set, which is I think how it works for a set like N.

      So I tried to draw a set with no elements (that red blob) at the very start of the picture, with infinitely many elements in the infinite set unrolling out as the set converges with the horizon. Infinite is there, but it’s building on that initial minimal element, that red empty set.

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  1. Isn’t regularity what keeps a set from being an element of itself? So each set has an element. I know it prevents infinite descent, too. But the causality confuses me. Each set has an element disjoint from the set?

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    1. I don’t think it claims that each set has an element — otherwise, the empty set couldn’t exist.

      Thanks for pushing on this. Let’s see if we can both get clearer on this! I looked back at my text and I think regularity directly implies both (a) that a set can’t be a member of itself and also (b) no infinite descent. I don’t think either result “causes” the other, though once you have no infinite descent you can also bar any set from being a member of itself (as “x is in x” would create an infinitely descending chain of “…x is in x is in x is in x”).

      So, how does the axiom prevent infinite descent? Suppose that you had one of these infinite chains. The example I keep on thinking about is the set containing just one person. That person is a set of biological systems. Each biological system is a set of organs. Each organ is a set of cells. Each cell a set of organelles. So on, and so on. Suppose that this went on forever.

      If you had one of these infinite chains, you could collect each set in the sequence into a NEW set. This set contains as elements: the person, the set of biological systems, the set of organs, the set of cells, the set of organelles, etc., etc.

      The axiom of foundation says: this set contains something whose elements aren’t elements of this NEW set. (In our example, maybe this “something” is the set of quarks or whatever is posited to be truly basic in particle theory. I have no clue.)

      Here’s the thing, though: if that something was part of the infinitely descending chain then it would HAVE to contain something further. Otherwise, it would stop the chain. It needs to contain some further set, and that further set would be part of that NEW set.

      I have no idea if that’s any clearer. I feel like it’s not as clear as it could be yet.

      Anyway, if there are no infinitely descending chains of sets then, as a corollary, no set can contain itself.

      (My textbook contains a proof that starts like this: “Let z be any set. By using the axiom of foundation with x = {z} show that z can’t be a member of itself.” That would show that you can’t be a member of yourself independently of the “no infinite descent” argument.)

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  2. Okay, I think I’m closer. This is only relevant to sets of sets. (That’s probably a duh.) My stopping point has always been, S={1,2,3…}, what element is disjoint to S? 2 intersect S isn’t even meaningful. And the empty set is not an element of S though it’s always a subset. But for sets of sets, OK. Now I’m thinking about the picture.

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    1. I don’t know if this will help, but from the perspective of set theory “2” actually *is* a set of sets. That’s because the natural numbers are defined entirely in terms of sets in the following way, typically:

      0 = empty set
      1 = { {empty set} }
      2 = { {empty set}, {{empty set}} }
      etc.

      So 2 intersect S is meaningful, given that definition, and there’s a sense in which 1 is a member of both the set S as well as a member of 2.

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    1. I don’t think this is necessarily always true, but I think the axiom makes most sense if we think only about this world of only sets, no non-set elements.

      (If I understand things properly, any non-set element would fulfill the requirements of this axiom in a set. But I don’t think you can have infinite descent + a non-set element, because then you could just define a subset as the complement of that non-set element in the set, and you’re back with an infinitely descending set.)

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