by Brian Bosse, Copyright March 15, 2008, all rights reserved. 110 views
Prove: R is not countable.
Assume: R is countable.
By the definition of Countability, if R is countable, then |R|≤|N|. Since N is a proper subset of R, then we know that |R| is not less than |N|. Therefore, if R is countable, then |R|=|N|.
We also know by the definition of same Cardinality that if |R|=|N|, then there exists a bijection between R and N. Specifically, there exists a function whose domain is N and whose range is R.
Domain (N) Range (R)
1 f(1) 256
2 f(2)
3 f(3)
4 f(4) 5/9
. . .
. . .
. . .
n f(n) r
. . .
. . .
. . .
What Cantor's proof will do is show that there exists an element of R that is not in the range of f. That is to say, in the above listing there is an r that is not listed. This contradicts the claim that f is bijective. Since we have reached a contradiction from the assumption that R is countable, then the conclusion is that R is not countable. In part II we will look at how Cantor does this.
-----No comments yet.