The Most Original Idea I Know

Jack Sparrow said it best.

Will Turner: This is either madness… or brilliance.

Jack Sparrow: It’s remarkable how often those two traits coincide.

I define original to mean both useful and least like other ideas. There’s plenty of unique but worthless ideas out there, and we’re going to ignore them. Just string random words together and you get something like “a book about a unicorn in a wheat field in Slovenia”.

So by those criteria, Georg Cantor’s ideas about the sizes of infinity are the most original ones that I know of.

They led, at least loosely, to the following ideas:

  • The idea that “counting” a set \(X\) is just finding a bijection \(f: X \to \mathbb{N}\)
  • A proper subset of a set \(X\) can have the same cardinality as the whole set
  • Not all infinities are the same size
  • Diagonalize attn arguments, which Godel’s theorem used
  • Godel’s incompleteness theorems
  • A proof that almost all functions are not computable
  • Isomorphisms in category of sets
  • influenced idea of isomorphism in general, a view picked up by category theory

To say “there are as many even numbers as even and odd numbers” sounds insane. But it’s true in a way that corresponds to intuition about counting and indicates that infinity is a strange thing.

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