### Maths, Just in Short Words

#### Posted by David Corfield

*Guest post by David Roberts*

How much maths can you talk about if you could just use short words? Late one night, somewhere between waking and sleeping, the cartoon proof of Löb’s theorem and Boolos’ explanation of Gödel’s second incompleteness theorem (Paywall! But the relevant part is all on the first page, if you can see that) teamed up to induce me to produce a proof of Cantor’s theorem of the existence of more than one infinite cardinal, using only words of one syllable (or, “just short words”). I then had the idea that it would be interesting to collect explanations or definitions or proofs that used only words of one syllable, and perhaps publish them in one go. Below the fold, I give my proof of Cantor’s theorem, comments and criticisms welcomed. I was aiming for as complete a proof as I could, whereas a number of other people (more on that below) went for simplicity, or as short as they could, so the style is different to the others I mention.

### The fact of Georg C., in short words.

- A “set” is made up of things.
- The set $\mathbb{N}$ is made up of $1,2,3,4,\ldots,n,\ldots$
- We can add 1 to a thing in $\mathbb{N}$ to get a new thing in $\mathbb{N}$, not the same as the first.
- A “map” from the set $A$ to the set $B$ is a rule that takes a thing in $A$ and gives a thing in $B$.
- For a set $C$ and a set $D$, the set $Map(C,D)$, “maps from $C$ to $D$”, is made up of
*all*maps from $C$ to $D$. - The set $A$ is said to be “not as big as” the set $B$ if: when you give me a map from $A$ to $B$, I can find a thing in $B$ that does not come, by the rule for the map, from a thing in $A$.
- Let M be a map from the set $A$ to the set $Map(A,\mathbb{N})$.
- This means that $M$ is a rule that gives a map from $A$ to $\mathbb{N}$, for each thing $a$ in $A$. We will call this “$a$’s Map”, with the fact this is by the rule for $M$ in the back of our minds.
Now I will tell you a thing $C$ in $Map(A,\mathbb{N})$. It is a map from $A$ to $\mathbb{N}$, with rule:

the thing $a$ (in $A$) goes to the thing that $a$’s Map sends $a$ to (a thing in $\mathbb{N}$) plus 1.

Read that once more, as it’s a bit hard to grasp.

- I claim that $C$ is a thing that does not come from, by the rule for $M$, from a thing in $A$.
- Why? Since if it did, let us say, come from the thing $a$ in $A$, then $C$ is $a$’s Map for some $a$, and so the thing to which $a$’s Map sends $a$ is the same thing to which $C$ sends $a$, which is the thing to which $a$’s Map sends $a$, now plus 1, which is false, by point 3. So $C$ is not $a$’s Map!
- So we were wrong to think that $C$ was a thing that came from $A$ (by the rule for $M$), and so $A$ must be not as big as $Map(A,\mathbb{N})$.
- If we take $A$ to be the set $\mathbb{N}$, this means that though $\mathbb{N}$ is a big, big set, it is not as big as $Map(\mathbb{N},\mathbb{N})$!

This proof I learned on the homotopy type theory mailing list (this post by Thomas Streicher), and I like it as it uses (almost) minimal logical assumptions: no need for power sets, Axiom of Choice, excluded middle etc. There is of course Lawvere’s proof via his fixed-point theorem, which doesn’t even require function spaces, though it is less clear what this means for cardinalities. To me, though, it feels like an a proof in ‘external logic’, reasoning about the ambient category of discourse from the outside. The proof above works in the internal logic of any $\Pi$-pretopos with natural number object, I believe. Probably it works if we replace the abstract object $A$ with $\mathbb{N}$ and assume that $\mathbb{N}^\mathbb{N}$ exists, rather than assuming cartesian closedness. But this is off-topic.

After I mentioned this on Google+, Joel Hamkins sent me a different proof of Cantor’s theorem, all in single-syllable words, then a proof of the irrationality of $\sqrt{2}$. Then Tim Gowers wrote another irrationality proof, and then, on my prompting, an explanation and outline of the proof of the Green-Tao theorem on arithmetic progressions in the primes. Toby Bartels wrote out the Peano axioms. Asaf Karagila wrote a proof of Cantor’s theorem without using the letter ‘e’. You can can a number of these in the comments at my original Google+ post, but read on before clicking through.

What I’d really like to see is two things. First: people who have proved amazing theorems explain said theorems in this manner (I was hoping to get, for instance, Terry Tao or Ben Green to write up the Green-Tao theorem). Or perhaps see how many Fields Medallists we could get to do this exercise, on the result/concept of their choice. Second, and this is a more local challenge to Café patrons: write about some category-theoretic concept using monosyllabic words. Otherwise, anyone can chip in with their favourite proof, fact, concept or definition in the comments! It is best to give it a go before reading others’ attempts, especially if you happen to have chosen the same thing to write about as someone else.

So get those pens or keys and go: write maths in just short words!

## Re: Maths, Just in Short Words

Six plus one is… hmm.