Quote:
Originally Posted by John Carter
Great thread. Cept!
Here's an interesting, and to me very disturbing, consequence of Cantor's transfinites. In short, there are uncountably many real numbers that will never be named. There are, in fact, so many of these unnamed numbers relative to the numbers we can name that if we had a dart with an infinitely sharp point (such that if we threw it at the number line, it would hit exactly one number) and then threw it at the real number line, the probability that it would hit a named number is exactly 0!
|
It's even worse than that! To make things simpler, and just look at the closed interval [0,1]. So, the probability of randomly picking a number in a sub-interval of [0,1] is merely the length of the interval (for example, the probability of picking a number in (1/2,3/4) would be 1/4). In general, the probability of picking a number in a given subset of [0,1] would be the
Lebesgue Measure of the subset.* It turns out that any countable set--in particular, the subset of rational numbers in [0,1]--has a Lebesgue measure of zero. Therefore, the probability of choosing a rational number is zero.
In fact, there are only countably many
algebraic numbers (ie numbers that are the root of a polynomial equation with integer coefficients), so the probability of picking an algebraic number in [0,1] is also zero. Numbers that are not algebraic are called transcendental. There are very few transcendental numbers that we've named (pi, e, a few others (along with their obvious sums and multiples)), but almost all numbers are transcendental.
* - Not every set is Lebesgue measurable if one accepts the
axiom of choice.