Tuesday 10 February 2015

Possibility of the Actually Infinite

Mathematicians define two sets as being the same size whenever there is a bijective mapping from one to the other. That is to say, whenever each member of the one set can be paired up with exactly one member of the other. At face value this makes perfect sense. If you're at a dinner party and every guest has brought a significant other of the opposite sex, then clearly there are just as many men at this party as there are women.

Why does this matter? Because, if same size is understood as mathematicians define it, then it easily follows that a set can be the same size as its proper subset. In other words, the common objection to the possibility of actual infinities simply fails by definition.

It's all the more clearer with an example. The natural numbers {1, 2, 3, ... } and the even numbers {2, 4, 6, ... } are the same size because a bijective mapping exists between them: f(x) = 2x, which pairs 1 with 2, and pairs 2 with 4, and 3 with 6 and so on. And, of course, the even numbers are a proper subset of the natural numbers.

Now you might think, this sounds super abstract. Maybe this is some weird technical idea that mathematicians throw around, but surely all this mathematical mumbo jumbo isn't anything like my understanding of what it means for two sets to be the same size. But actually, it is.

When someone wants to know how big a set is, they count the members. They point to an object and say "one", they point at another and say "two", and so on. They don't know it, but they're proving the existence of a bijective mapping between the set in question and the subset of the naturals they're vocally describing. And because the size of the set they're counting out is identical to the value of the last member, they (all by intuition and not understanding the math) infer the size of the set in question. It's odd to think that something so simple as counting has a rigorous mathematical basis that utilizes the mathematicians very technical definition of same size, but it in fact does.

Of course we needn't proceed one member at a time when counting. We could just as well point to a pair and say "two", and another pair and say "four", and so on. Or we could go by groups of ten, or hundreds or, even, the collection as a whole. It follows from this that, contrary to popular opinion, we can count an infinite set after all, we just do it all at once instead of step by step.

Here's the catch, then. People who think that actual infinities are impossible (almost always because they think it's impossible for a set to be the same size as its proper subset) owe the rest of us an explanation of what 'same size' means, if not what mathematicians mean. Since they are unable to give any answer, their objections to actual infinities (usually to the possibility of an eternal past) fall flat.