The set of all singletons doesn’t exist

From Classic Set Theory:

Use appropriate axioms out of Zermelo-Franekel axioms 1-6 to show that \{x: x \textup{ is a singleton} \} is not a set.

I’m fascinated by sets that go wrong, and I was surprised that set of all singletons leads to trouble. Here’s my best attempt at explaining what goes wrong.

The issue with “is a singleton” as the defining property of a set is that any set can be turned into a singleton via the Axiom of Pairing in ZF set theory. The Axiom of Pairing is like a machine: give it two sets, it’ll smoosh them together into one set. Give it the same set twice, and it produces a singleton. That way, there’s a singleton for every set.

Slightly more rigorously, let X be any set at all:

“Hi, my name is X and I’m a set.”

Feed two copies of X into the Axiom of Pairing; the axiom then spits out the set whose elements are just the elements of X and, well, X:

“Yo. The Axiom of Pairings made me. I’m the set whose elements are the same as those of X and … err, ok, so just X. I’m \{ X \}.

So, for any set X there exists a singleton set whose only element is X. Great! So what’s wrong with the set of all those singleton sets?

The problem is another axiom, the Union Axiom. The Union Axiom is another little set theoretic machine. Feed the Union Axiom a set (chomp chomp) and it spits out another set. This new set contains all the elements that are in subsets of the original.

An example is probably useful. Say that your set is the set of all the basketball teams in the NBA: {Bulls, Pacers, Knicks, etc.}. The Union Axiom is the set of all the players in the NBA. The Union Axiom bites through the husk of one collection and produces a new one out of the things living just one level down.

Here’s where we start breaking things: what if you feed the set of all singletons to the Union Axiom? (Get ready, because here comes everything.)

  1. The Axiom of Pairings can turn every set into a singleton.
  2. The set of all singletons collects all of these singletons into a set.
  3. The Union Axiom would create a collection of all the elements of elements of the set of all singletons, i.e. literally every set.
  4. So the set of all sets exists.

And that’s great, if you love contradictions, because now you can make any set you want, including Russell’s famous one. Because in this version of set theory the way that you block Russell’s Paradox is with the Axiom of Separation, which says that you can define a set using any property, as long as it’s a subset of some currently existing set. Now, though, we have a set of all sets. Everything can exist, bats explode out of the belfry, Pandora breaks the seal, boom!

“Let there be a set that’s a subset of that set of all sets, containing all of the sets that are not members of themselves.”

And that’s Russell’s paradox, and it leads to contradiction.

All of which is a long way of saying, the set of all singletons does not exist. Which is weird, because I had written previous that this was the way Frege defined “1” within his system, as the set of all singletons. Is this set itself one way that the problems with Frege’s system manifested itself? Or was I misunderstanding Frege’s definition?

One subtlety I’m missing out on right now is how the Zermelo-Fraenkel set theory system compares to other set theories. This bit from Wikipedia seems to summarize what I’ve just started to understand:

Gottlob Frege and Bertrand Russell each proposed defining a natural number n as the collection of all sets with n elements. More formally, a natural number is an equivalence class of finite sets under the equivalence relation of equinumerosity. This definition may appear circular, but it is not, because equinumerosity can be defined in alternate ways, such as by Hume’s principle.

This definition works in naive set theory, type theory, and in set theories that grew out of type theory, such as New Foundations and related systems. But it does not work in the axiomatic set theoryZFC and related systems, because in such systems the equivalence classes under equinumerosity are proper classes rather than sets.

I’d like to understand some of the other sets that lead to trouble in ZF set theory. What sort of properties, in general, lead to contradiction? And I’d also like to understand what sort of choices you make in type theory in order to permit this sort of definition of “1.”

Gutierrez on the tensions of teaching

Although teachers must recognize they are teaching more than just mathematics, they also have to reconcile that fact with the idea that, ultimately,  they are responsible for helping students learn mathematics. Teachers who are committed to equity cannot concern themselves with their students’ self-esteem and negotiated identities to the exclusion of the mathematics that the students will be held responsible for in later years. Yet preparation for the next level of mathematics must also not be the overriding feature of a teacher’s practice. In answer to which of the two foci are important (teaching students or teaching mathematics), I would answer “neither and both.” It is in embracing the tension…”

Rochelle Gutierrez, Embracing the Inherent Tensions in Teaching Mathematics from an Equity Stance

What does it mean to live in tension? I suppose it’s an emotional feeling, of not knowing at any given moment which way you should lean. I like the idea that teaching involves tensions that can’t be resolved, though I admit I’m still fuzzy on what it means to teach in a way that reflects such a tension.

“1” and Russell’s Paradox

I’m falling in love with Classic Set Theory, an introductory set theory text. The introduction to Russell’s paradox motivates it in a way that I’d never seen before.

When I’ve thought about Russell’s paradox in the past I’ve sometimes thought, ok, this makes sense, but how did he come up with this? Or, even, what line of thought could even have led to conceiving of something so clever?

(Russell’s paradox is one of the great stories of math: a letter to Frege, just as he’s completing his master work, announcing a fatal contradiction in his system, effectively ending Frege’s life’s project. Wow! The above panel is from Logicomix, a book that I left feeling disappointed by. I felt it didn’t live up how awesome the story is, but maybe my expectations were just too high.)

Anyway, here’s what I think I currently understand:

Frege figured out a way to define “1” without talking about oneness. Instead, he made it all depend on uniqueness.

Definition: A set C is a singleton if there is an x \in C and if anything else is in C, that thing is equal to x.

Frege doesn’t say anything about the number “1” in defining singletons; it all depends on equality. So Frege is free to use the notion of singleton-ness to define the number “1”.

Definition: There is a set of all the sets that are singletons. Call that set [1].

It’s a pretty nifty trick, and it shows that lurking within our notion of equality is some notion of oneness.

So far, that’s Frege: “1” is a set, [1].

But if [1] is a set, we can start playing around with that set. For example, can make a new set that contains just one element:

\{ [1] \} is a singleton!

Which leads to the weird notion that { [1] } is a member of [1]. And, of course, [1] is in ${ [1] }$ itself, so we get a weird series of inclusions:

[1] \in \{ [1] \} \in [1]

Not paradoxical yet, but weird. And close enough to Russell’s paradox that you can see where he might have started playing around with these ideas. OK, so there’s a weird way that [1] as defined by Frege is a member of itself. How could we mess around with that idea some more?

Definition: V is the set of all sets.

And then clearly, V is also a member of itself.

And now we’re so, so, so close to Russell’s paradox itself: the set of all sets that doesn’t include itself.

Part of what I find so exciting about all of this is the connection to Frege, but also the sense that there’s a natural progression from the definition of “1” to the paradox. What it is is that the definition of “1” does open the door to weird inclusions, since it’s a set of everything that is a singleton. And once that door is open, everything can march on through.

The only way to close that door is to restrict what is allowed to become a set.

In response to Heidi’s post

https://tooteeter.blogspot.com/2017/12/8-12-by-12-inch-paper-cardinal-vs.html

How did you get into my 3rd Grade classroom?? I was dealing with all of this just a few weeks ago.

My situation was slightly different, and because it informs my thoughts about your questions, I”ll share it here.

I decided to move to measurement out of frustration with my students’ ability to subtract. In particular, I was asking them to find the distance between two numbers on the number line, and it felt like we weren’t going anywhere.

Based on previous years memories of students who start measuring at the 1′ mark, I began class with a picture of a ruler and a pencil on the board. I placed the pencil so one end was at the 1′ mark, the other at the 6′ mark. How long is this pencil, I asked the class? The first view was, as you might expect, 6′, and I don’t remember if another kid or I was the one to disagree, but I drew the class’ attention to the alternate view, which was based on two things:

a) Counting the spaces, not the lines
b) Our ability to slide the pencil to anywhere that we want on the ruler (i.e. it wouldn’t make sense for the length to change)

Then I gave out rulers and asked them to measure that piece of paper, that same Investigations activity.

OK, and one last thing: when kids end up with different measurements, one thing I’ve learned to do is to turn it into a statistics problem. At the end of their measurements I tallied their measurements on the board in a list. I point out that variation in measurement is totally normal — there’s no way to make a perfect measurement, there is always error. So, I asked, looking at our measurements, what do we figure the TRUE measurements of the paper to be?

What does all this mean? My first guess is that a lot of number line strategies don’t make a ton of sense to kids yet, but that measuring helps. When we want kids to realize that 20 + 30 = 15 + 25 because you can slide both down the number line? That seems like a strategy best built on something like experiences with a ruler.

(It also reminds me of this research paper that we talked about, where teachers who focused more on measurement seemed to help struggling students more than those teacher who dug in on arithmetic.)

Second, an alternative to asking kids to remeasure until they get the correct measurement might be to record their various measurements and then to ask the statistical question about them: what do we think the true measurement is, given the spread of measurements we’ve been making? can we remeasure to get closer to that true measurement? Because, unlike addition or subtraction, there isn’t a way to measure the true length — every measurement is an approximation.

All this said, I’m not really sure why it makes sense to kids to start at 1 instead of 0. I like your idea that moving between tiles and lengths might get at the true relationship, but I find myself still puzzled about how kids see the ruler.

Rough Draft Talk, or the Avalanche

The phrase belongs to Amanda Jansen. I love the way “rough draft talk” sounds, and I think about it all the time.

There’s something a bit selfish, I think, about making other people listen to you work out your ideas all the time. It’s something I’m incredibly self-conscious about.

This is probably part of why the internet stresses me out so much. I’m a guy that just wants to talk to other people and work our thinking out, but too often this (feels like it) comes off as ranting. And it certainly is in tension with a desire to show the results of this thinking in something like a finished form. But should my blog be a place to publish thoughts or to work it all out? And if people read my blog because they like the finished thoughts, is it fair to subject them to everything else?

My answer is no, so I needed another online space to muck around in.

By the way, I very badly wanted to name this blog The Avalanche after the album of overflow material from Sufjan Steven’s Illinois. I decided that was presumptive and not exactly what I was going for, but that’s sort of what I want to do in this space: everything else.