## Cantor’s Theorem

Posted: June 16, 2008 in Fly-By-Night
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Some infinities are bigger than others.

Literally.

Imagine the top left corner of an infinite grid square. Infinite rows begin at the left-hand edge: infinite in length, and infinite in number.

Each row contains a unique arrangement of zeroes and ones – infinitely diverse combinations.

The first number of the first row nestles in the corner of the grid. It is also the first number on the diagonal, cutting evenly between the top of the grid and the left-hand edge.

Descending along this axis, imagine the reverse of every number. Zeroes become ones. Ones become zeroes. Completely opposite.

Make this alternate diagonal a horizontal row, and set it above the grid. Does it have a twin in the infinite rows?

No. You have just created an entirely new combination: one which cannot be found in the infinite sets.

If you tried to match up your new row, you would find that one digit is always reversed. If the grid begins with a zero, your new row begins with a one. If the second digit in the second row is a one, your second digit must be zero. This pattern holds no matter how far down, or how far across, you check, because your row is composed entirely of changes to the original, infinite, rows.

Which means, put simply, that your new row now exists in a bigger infinity.

Nifty, huh?