# Zero Sum Game is on sale! Also, some math for you about socks.

In celebration of the release of *Root of Unity *(less than a week away, w00t!), today marks the first day of a 99-cent sale of the ebook for *Zero Sum Game!*

**Amazon * Amazon UK * Barnes & Noble * Kobo * Apple**

In honor of this sale and so I’m just not Promoty McPromotion, I’m going to tell you about the title of the series.

*Russell’s Attic *may not be the best title for a book series about violent superpowered mathematicians — it doesn’t quite speak to the superhero aspect, or suggest adrenaline-pumping thrillers, or . . . anything else a title is supposed to do. But it was too good to pass up.

You see, Russell’s attic is an actual thing. I don’t mean a physical thing, but an actual mathematical metaphor.

Bertrand Russell is a famous mathematician and set theorist (yes, I named my main character after him, trufax!). He proposed the following thought experiment:

Say you have an attic filled with countably infinite pairs of shoes and countably infinite pairs of socks. (“Countably infinite” essentially means there’s a way to write them all down in a list — the list can be infinite; we just have to be sure it contains all the elements. For example, we are confident the list 1, 2, 3, 4, 5… contains all the natural numbers, even though it’s infinite. Similarly, we are confident the list 2, 4, 6, 8… contains all the even natural numbers, even though it’s infinite. So these are both countably infinite sets. The real numbers — think all possible decimals — are *un*countably infinite, because there’s no way to list them all, even in an infinite list.)

So in Russell’s attic, we have countably infinite pairs of shoes and countably infinite pairs of socks. The shoes in a pair can be differentiated from each other, left versus right. The socks in a pair are identical.

We know there are countably infinite *pairs* of both, and that each pair has two elements. The question: can we prove there are countably many *shoes* (not pairs of shoes) and countably many *socks* (not pairs of socks)?

The shoes are easy! We know there are countably many *pairs*, so there must exist a list like this:

{ Shoe Pair 1, Shoe Pair 2, Shoe Pair 3, Shoe Pair 4, Shoe Pair 5 … }

To make our list of all the *shoes,* we just do this:

{ Left shoe from Pair 1, right shoe from Pair 1, left shoe from Pair 2, right shoe from Pair 2, left shoe from Pair 3 … }

Since we’ve already said the first list exists, the second must as well, and we have countably many shoes. Presto.

Now let’s try to prove there are countably many socks. If we know there’s a way to list the *pairs* of socks, is there a way to list the *individual* socks like we did for the shoes?

…

…

…

Um.

It turns out it is impossible to prove there are countably many socks *unless you use the Axiom of Choice. *Even though we can easily prove countably many pairs of shoes means countably many shoes in the attic, and we can do it *without* the choice axiom, we can’t do the same when we start with countably many pairs of socks.

Whoa.

That punchline may be a little anticlimactic if you don’t know what the Axiom of Choice is. So what is it? Basically, the Axiom of Choice says that if you have a collection of sets of things, it is possible to grab one thing out of each of the sets.

Sounds obvious, right? It did to mathematicians, too, who for a long time simply assumed this was true for any collection of sets because of course you can do that. Then a guy named Zermelo came along and showed that assuming this *super obvious thing* led to a result that *blew mathematicians’ minds.*

The mind-blowing result he proved is called the well-ordering theorem, and from what I’m told it caused a minor apocalypse in the mathematical world, because *it was so obviously wrong how could it possibly be true.* So Zermelo went back through his proof and showed that the only assumption he’d made was that you could pick one thing out of each set of things, which of course everyone accepted as obviously true. But he’d used this Super Obviously True thing to prove something everyone knew was Obviously Impossible.

Thanks to Zermelo, we now know the obvious thing and the impossible thing are actually *the same thing*.

*(the three are equivalent)*

So the formerly-super-obviously-true thing became a formal axiom instead, called the Axiom of Choice. And without it, you can’t prove there are countably infinite socks in Russell’s attic.

As for my book series title, since “attic” can also imply someone’s mind or give a metaphor for their general state of being, I thought *Russell’s Attic* was perfect!

Interestingly, I just did a search for “russell’s attic” (without the quotes), and the first result is the Dictionary.com definition of the mathematical thing… and every other first-page result is now this series.

Oops. Sorry, math world!

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