Shatar: You know what I’ve always wanted to do?
JJ: Breath? Eat food? I suspect that these are some of the few desires that you have always held.
Shatar: Well I already had eggs and pidgeon for breakfast, they were really tasty. So that need satisfied now what I want to do is add up a bunch of numbers.
JJ: That’s actually surprisingly healthy, I’m impressed. And adding up numbers is certainly a typical activity in mathematics. In fact, recently I have been exploring some interesting additive combinatorics that I could tell you about, relating to the set of possible subset sums–
Shatar: Nah, that’s adding up just a finite amount of numbers, I want more than that!
JJ: Ah, well, maybe you could check out some of my calculus tutorials, we routinely take limits of finite sums, I think you –
Shatar: No, that’s not enough numbers either! I want to do more.
JJ: Oh dear.

Shatar: ok, so we all know that if you add up a countably infinite amount of numbers then you can get some pretty interesting things, like

\[\sum_{n=1}^{\infty}\frac{1}{n^{2}} = \frac{\pi^2}{6}.\] But what if you added up countably infinitely many collections of countably infinitely many things?

JJ: I’m affraid that the union of countably many countable sets is itself still countable by a simple diagonalization argument:

\[\mathbb{N}^{2}\cong \bigcup_{i,j}X_{i,j}.\]

Shatar: ah, rats!

Shatar: Ok, that means we need to go bigger. I bet I get something pretty interesting if I take a set \(X\subset [0,1]\) which is uncountably large and try to add them all up.

JJ: Hmm, I think it will not be quite so interesting as you think.

Shatar: Oh, yeah? Well let’s make it interesting!

JJ: An admirable sentiment my friend, but ultimately futile.

Shatar: How so?

JJ: Imagine that you had an uncountably infinite number of pidgeon omletes–

Shatar: :O so tasty

JJ: – and put them into countably many frying pans. Then one frying pan would necessarily have uncountably many pidgeons in it. This is due to the lemma we proved above about the union of countably many countable sets being countable.

Shatar: Mmmm! Yum.

JJ: Now partition \([0,1]\) geometrically into intervals \((\frac{1}{2^{i+1}}, \frac{1}{2^{i}}]\) for \(i\in 0,1,2,\ldots\). One of the intervals must have infinite intersection with \(X\), call this interval \(I=(\frac{1}{2^{i_0}}, \frac{1}{2^{i_0-1}}]\). But then the sum of the elements of \(X\) that lie in \(I\) alone is infinite. So \(\sum X = \infty\).

Shatar: hmm. well. that’s quite a lot.

JJ: Indeed.

Shatar: Thanks for the help!

JJ: Did you eat my pidgeon omlete?

Shatar: Um… Good thing we had an infinite number of pidgeons right?