Infinities

Infinity is a funny thing and sometimes the infinity of the small surpasses the infinity of the large.

Take rational numbers, for instance. A rational number is any number that can be expressed as a ratio of two integers. There are an infinite number of them. In fact, between any two rational numbers, say 1 and 101/100, there are an infinite number more. But just how infinite are they? I could bore you with a proof, but this post about philosophy rather than mathematics, so just take this as a given:

The set of natural numbers and the set of rational numbers are infinite to the same extent.

Both are considered "countable." You can mathematically map the set of rational numbers onto the set of natural numbers. Try it for fun and profit!

What about real numbers? A real number can be thought of as a representation of a point on a line where the integers occur at regular spacing and every real number represents the distance from an (arbitrary) origin that represents zero. Cantor proved that the set of all real numbers is "uncountable", i.e. it is mo' better infinite than the set of natural numbers (and rational numbers).

All rational numbers are also real numbers, but not all real numbers are rational. You can easily construct a line that is the square root of 2 long (the hypotenuse of a right triangle with sides 1 unit long each) and then prove that the square root of 2 is not rational. An irrational number is defined as any real number that is not rational.

Now comes the kicker - The real number set is uncountable, and the rational number set is countable. If you remove a countable subset from and uncountable set, the remainder is still uncountable and so the set of irrational number is uncountable. The set of irrational numbers is mo' better infinite than the set of rational numbers. OK - So what?

So what is this. On that continuum of real numbers, it is not difficult to contruct a rational number or an irrational number between any two arbitrary point, so you can show that:

1. Between any two rational numbers, there is an irrational number; and


2. Between and two irrational numbers, there is a rational number.

But the set of irrational numbers is uncountable, and there are infinitely more irrational numbers than rational numbers!