7:26 AM — Infinite Monkeys throw Infinite Shit at Infinite Computers Monkeys can't write Shakespeare. They can't even figure out where the keyboard is on a computer. And even if they could find the keyboard, typing an infinite number of letters for an infinite time is still no garuntee. For example, they could simply type an infinite number of letter A's. That would take an infinite time, yet they would never get the complete works of Shakespeare, abridged or no. Gotta love infinities.

• There's an infinite amount of positive integers (at times also called counting numbers, whole numbers, or natural numbers, though these terms can be confused with the next category). These numbers can be listed: 1, 2, 3...


• There's one more non-negative integer: 0, 1, 2, 3... So that's infinity plus one (equals infinity). Alternately, you can think if mapping all the positive integers (p) to the non-negatives (n) by saying that n=p-1, and therefore there's actually the same number of infinite values in each set. This is referred to as the order or cardinality of the set, which is called alef-null.


• There's twice as many plus one integers as positive integers, or twice as many minus one integers as non-negative integers, all of which are infinite: ...-3, -2, -1, 0, 1, 2, 3...


• There's even more infinite rational numbers, anything that can be expressed as a fraction n/m where n and m are both integers, for example (in no order and establishing no trend) -1/2, 25/342, -7, 44/3...


• And there's infinitely more yet of real numbers, consisting of both rational and irrational numbers, numbers that can be expressed as a fraction and those that can't. For example, irrational numbers include sqrt(2), pi, -(4/3)^(1/5)... This is a higher "order" (alef-one) of infinity than the previous ones (all alef-null). This one's continuous (no "gaps" between numbers, can completely fill the "space") while the previous ones are discrete.
  

References
Cardinal Number (c/o MathWorld) discusses the levels of infinity.
Counting Numbers (also MathWorld) defines the different types of "whole" numbers.
Monkeys Don't Write Shakespeare (original AP article c/o Wired News)
Notes Towards the Complete Works of Shakespeare (official page of the experiment/performance art)
White Light by Rudy Rucker, a surreal sci-fi novel regarding infinities and God.  ¶ permalink | 0 musings