## Interesting / Fun on Cantor’s Diagonalization Proof

December 15, 2009 by un1crom

I really like this post on Good Math, Bad Math.

Beyond being mildly humorous in that cranky math person non-funny kinda way, it touches on lots of my favorite subjects: enumeration, Cantor, classic proofs, cranky math people.

The catch – and it’s a *huge* catch – is that the tree defines a *representation*, not an enumeration or mapping. As a representation, taken to infinity, it includes every possible real number. But that doesn’t mean that there’s a one-to-one correspondence between the natural numbers and the real numbers. There’s no one-to-one correspondence between the natural numbers and the nodes of this infinite tree. It doesn’t escape Cantor’s diagonalization. It just replaces “real number” with “node of this infinite tree”. The infinite tree contains uncountably many values – there’s a one-to-one correspondence between nodes of the infi To see the distinction, let’s look at it as an enumeration. In an enumeration of a set, there will be *some* finite point in time at which any member of the set will be emitted by the enumeration. So when will you get to 1/3rd, which has no finite representation as a base-10 decimal? When will you get to π?

FUN!

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