Talk:Countable set

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[edit] Countable rationals

There are lesser-known sequences for counting the rationals other than Cantor's mapping. For example, the following sequence assigns each natural to a unique positive rational:

S(0) = 0
S(1) = 1
S(2n) = S(n) + 1 for all n > 0
S(2n+1) = 1/S(2n) for all n > 0

This sequence is based on Farey sequences and Stern-Brocot trees. It can be extended to cover the negative rationals as well. — Loadmaster (talk) 17:40, 3 February 2008 (UTC)

[edit] Cantor: A Mathematical Charlatan

Consider the set of natural numbers listed in the usual order. That is, starting with 1,2,3... and written from left to right. By Cantorian definition, this set is both "countable" and infinite. Therefore we should be able to place the elements of this set in 1 to 1 correspondence with the set of natural numbers listed in any other order. Let us choose to use 'reverse order' for this second set. Now suppose we place the first set above the second and try to match elements. Then we should find one and only one entry under the number "1" in the first set. What is this number? Clearly, there is no such number! The concept of "countably infinite sets" is inherently flawed. This concept is the basis of the pseudo-mathematics of Cantor and his followers.

What can be counted can not possibly be infinite, and what is infinite can not be counted. Karl Freidrich Gauss pointed this out when he wrote "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics". The flaw in the 'diagonal argument' has precisely this defect. —Preceding unsigned comment added by 66.67.96.142 (talk) 22:16, 23 May 2008 (UTC)

Sets do not distinguish order. The set of the natural numbers in any other order is the exact same thing as the set of the natural numbers in their usual order. Therefore the bijection is trivial. 195.197.240.134 (talk) 11:08, 28 May 2008 (UTC)