Re: cantor's theorem - the pieces of the puzzle
- From: Virgil <Virgil@xxxxxxx>
- Date: Fri, 16 Nov 2007 12:34:05 -0700
kunzmilan wrote:
One fault of these discussions is, that either contributors do
not follow them, or they forget what was discussed and solved.
1) The set of rationals 1/k, k going from 1 till infinity is
countable. It is infinite. This set does not contain all rational
numbers. Thus rational numbers can not be counted using the set of
natural numbers.
The set of ALL rational numbers has quite frequently been ennumerated by
the naturals.
2) The set of rationals j/i is is countable, and their counting
function exists. But it grows faster than n. Thus rational numbers
can not be counted using the set of natural numbers.
Consider the mapping of rationals to naturals:
For q = m/n, with n a positive natural, m an integer and m and n
having no integer common factors greater than 1,
if m < 0 map m/n to 2^(-m)*3^n, or
if m > 0 map m/n to 5^m*7^n
Then every rational is mapped to a different natural, so there re at
least as many naturals as rationals.
These arguments are quite simple. I also tried to disprove Cantor.
And failed.
.
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