Infinite Numbers

ℵ0 (Aleph zero, null or naught) is the smallest infinity, the size of the natural numbers. It is countably infinite, which means there exists some method of counting that will eventually reach each item in the set.

The rational numbers – all fractions of the form p/q where p and q are natural numbers, is also of size ℵ0. One way to prove this is to demonstrate a method for pairing each natural number with a rational number, and show that every rational number will have a pair. The classic proof draws a table of rational numbers and walks through it starting in a corner and marching along diagonals.

So we have the intuitive result that you can pair off the elements of 2 sets with each other, if and only if the sets are the same size.

To me it seems counterintuitive that the rational numbers are the same size as the natural numbers, even though this fact follows logically from the very simple and intuitive above proposition. It seems like there are a lot more rational numbers. However, what follows seems even stranger to me.

The irrational numbers – π, e, and myriad others, are more numerous. Their size is a bigger infinity called ℵ1. They are uncountable – which means there doesn’t exist any method of counting that will reach all of them. Every method you come up with will skip some. There is no way to pair them off with the rational or natural numbers – no matter how you do it, there will always be irrational numbers left over without a pair.

Despite being countable, the rational numbers are infinitely dense. Between any two of them lie infinitely many more. The irrational numbers are also infinitely dense. What is more, between any two rational numbers lie infinitely many irrational numbers. But we’d expect that, given there are more irrational numbers. Furthermore, and most strangely, between any 2 irrational numbers lie infinitely many rational numbers. How can that be, if irrationals outnumber rationals?

The proof is simple. Pick any two irrational numbers, n1 and n2. Take the absolute value of their difference, d = | n1 – n2 |. There are infinitely many irrational numbers smaller than d. If that’s not obvious, pick some natural number ε greater than both d and 1/d. Then 1/ε is a rational number smaller than d.

It seems strange that 2 sets, each infinitely dense, both in itself and in each other, can be of different sizes. But they’re both infinite,so this is probably just a manifestation of the intuitive difficulty conceptualizing different sizes of infinity.