From e23e39442d4dd523535795a8d8a7ba25e3fa5ab3 Mon Sep 17 00:00:00 2001 From: joshuab <> Date: Sun, 11 Jul 2010 19:56:49 +0000 Subject: center --- ClassJune28.page | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ClassJune28.page b/ClassJune28.page index 40799dc..1987c97 100644 --- a/ClassJune28.page +++ b/ClassJune28.page @@ -16,7 +16,7 @@ Infinity, of course. So to perform a meaningful check that the set of nice perio For example, the open interval $(-\pi,\pi)$ has the same cardinality as $\mathbb R$. To see this, take the interval and bend it into a semicircle, then balance it on the real line. To find the point on the real line corresponding to a given number in the interval/on the semicircle, $\theta \in (-\pi,\pi)$, draw the straight line emanating from the centre of the circle at an angle $\theta$ with the vertical; it hits $\theta$ on the semicircle and its partner $x$ on the real line. Explicitly, the map takes $\theta \mapsto \tan \theta$. -![alt text](/stereographic circle.jpg) +
![](/stereographic circle.jpg)
There are, however, fewer natural numbers in $\mathbb N = \{1,2,3,\dots\}$ than there are real numbers in $\mathbb R$. Suppose to the contrary that it is possible to list the real numbers, or even just those in the interval $(0,1)$. Write that list down in binary, forming a big block extending infinitely downward (each row is a number), and to the right (the binary digits of each number). Read off the digits on the diagonal to get a number $x=0.010100110101110101011\dots$. Now toggle every digit of that number to get a new number $y$, say $0.101011001010001010100\dots$. This number $y$ is not anywhere on the list--it disagrees with the $n^{th}$ number on the list in its $n^{th}$ digit. Hence the list was incomplete. -- cgit v1.2.3