Bigger Than Infinity

"Give a man a fish, feed him for a day. Teach a man Cantor's Diagonal Proof, blow his mind out of his asshole. Now you don't have to feed him." --Philosophy Bro

Richard Brown mentions Cantor's Diagonal Proof towards the end of Episode 3 of the SpaceTimeMind podcast. This may perhaps be one of those things that are a bit hard to follow in a strictly audio format. Here's my own take on the proof. It's excerpted from "Welcome To Infinity," a short paper I wrote in 2012 for students in my class on the philosophy of science. The proof comes in the paper's final section, called "There's more!"

Read it, and we won't have to feed you.

An infinite set that has members that can be mapped one-to-one onto the natural numbers is a countable set. Georg Cantor proved the existence of infinite sets larger than this. Such sets are uncountable.  Cantor proved that there are more real numbers than integers.

If for each of the natural numbers we had a row upon which we wrote the infinite decimal expansion of a real number, then we could discover a number that is not on any of those rows by taking the diagonal (the number whose first digit is the first digit of the first row, second digit is the second digit of the second row, and so on) and changing each of the diagonal’s digits. The resultant changed diagonal is guaranteed not to have been anywhere in the original list since any number on the original list would differ from the changed diagonal at the digit where that number’s row intersected the diagonal. 

In the following example the digits in bold italic form the diagonal.

Row 0: 0.1234567…

Row 1: 0.2468101…

Row 2: 0.481632…

Row 3: 0.5101520…

Row 4: 0.98989898…

Row 5: 0.75757575…

Thus the diagonal number is 0.28087…The changed diagonal is a number that differs from Row 0 in having a different 1st digit, differs from Row 1 in having a different 2nd digit, differs from Row 2 in having a different 3rd digit, and so on. The changed diagonal will differ from every number on a Row by one digit.  Thus, the changed diagonal will not itself be one of the Row numbers. Thus the changed diagonal will not be mapped onto one of the natural numbers. AND... there thus exists at least one more real number (the changed diagonal) than there are natural numbers. Ta da!