# 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.”