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authorluccul <luccul@gmail.com>2010-07-06 04:12:13 +0000
committerbnewbold <bnewbold@adelie.robocracy.org>2010-07-06 04:12:13 +0000
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downloadafterklein-wiki-0c19db3ab7fcce14941d69fa589881a6a8dfafb4.tar.gz
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@@ -101,7 +101,7 @@ Proofs:
-Josh explained Cantor's proof of the uncountability of the real numbers on the 28th; Wikipedia provides a good description thereof: [external](http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument). In a nutshell, you assume you can have an ordered list of the reals (i.e., a bijection to the naturals), then construct a real number not on that list by having its nth digit be different from the nth digit of the nth number on the list.
--$(0,1)=\mathbf{R}$ and $[0,1]=\mathbf{R}$ under the same bijection: $n \mapsto \tan{\pi n - \pi/2}$.
+-$(0,1)=\mathbf{R}$ and $[0,1]=\mathbf{R}$ under the same bijection: $n \mapsto \tan\left (\pi n - \pi/2 \right)$.
-$2^{\mathbb{N}} = \mathbf{R}$ by writing a given real number $r$ as a (possibly infinite) set of natural numbers. For example, write $pi$ as the set of rational numbers $\{3,0.1,0.04,0.001,0.005,...\}$, then replace each number with the natural number it would map to under the bijection $\mathbb{Q}=\mathbf{N}$.