I have blogged before on infinity, which holds a certain fascination for me. For one thing, I’m working on a thesis involving self-reference and the retreat into infinite recursion. And early in my Presbyterian life, I had to defend divine sovereignty and the infinities of power, knowledge and goodness from the inroads of Arminian rationalism and free will. So with a little practice, I’ve gotten quite comfortable with infinity, sort of like driving 80mph in the dark in a thunderstorm–the important thing is not to think about it too long. In this post, I want to think about it long enough to show that the Multiverse doesn’t save Darwin.
Georg Cantor, of course, couldn’t stop thinking about it and was driven mad. But before he went into the sanatorium, he produced a most remarkable result about sizes of infinity. Some infinities are bigger than others. For example, take the number line from 1 to ∞. It’s infinite of course. But now divide every number by the largest number on the line, and we have mapped the entire number line into the fractions between 0 and 1. So the rational numbers contain the entire integer number line between 0 and 1, and the rational numbers go up to infinity too. Then the rational numbers are at least ∞2 bigger. (Yup, I’m being sloppy, because Cantor also showed how to map x2–>x, so instead of calling it ∞2, he called it ℵ0 cardinality where integers and rational numbers have the same size infinity.)