Monday, January 2, 2012

The Axiom of Choice

I can't think of a single good reason to blog about this, except that it makes me happy. That's good enough, right?

So, axioms. Axioms, if you didn't know, are the basic statements that we accept without proof. Logically, maths starts with a handful of axioms which are used to prove everything else. Well, almost. Kurt Godel showed that some things can neither be proved or disproved, which is where things get interesting. No matter what axioms you start with, there will either be inconsistencies, or ideas that can't be shown to be true or false.

Mathematicians have used different sets of axioms over the course of history, getting more and more precise. (Round and edible is not an incorrect definition of an orange, but it describes an apple too; round, edible and citrus is better, but still includes lemons.) The system that's most often used currently used is the creation of mathematicians Zermelo and Fraenkel. Their original system is abbreviated ZF, but the one used now is called ZFC. That's Zermelo-Fraenkel plus the axiom of choice.

The axiom of the choice is one of those things that can't be proved either way using ZF (some very smart people did the work to show that it can't be shown to be true or untrue), and it's been added to the basic set of axioms, like I added 'citrus' to my list of things that describe an orange. I like seeing how maths grows like that. It's a simple enough idea: it says that if I have a bunch of identical things, I can pick one without specifying which one to pick. It seems intuitive enough, but it has some weird consequences.

Particularly, it leads to the Banach-Tarski theorem. The Banach-Tarski theorem is so weird that it's usually called the Banach-Tarski paradox. It says that if you have a ball, you can chop it up into a finite number of pieces and then reassemble those piece to form two balls, each the size of the original.

See? Weird. Despite that, the axiom of choice has survived controversy to become the kind of axiom that is assumed to be assumed. And that is the power of sheer awesome at work in a mathematics near you.

Also, there's a band called Axiom of Choice. That's cool.

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