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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)
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. Carl Friedrich 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)
- It doesn't matter whether there is any way of arranging the elements so that they do not form a correspondence. All that matters is that there exists even one way or more to arrange them so that there is a one-to-one correspondence between the sets. And for the infinite sets there are infinitely many different ways of arranging the elements so that they correspond.
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- "What is infinite cannot be counted" -- not so either. Countably infinite is defined as having a bijection to the set of natural numbers. —Preceding unsigned comment added by 41.241.23.249 (talk) 15:19, 2 September 2008 (UTC)
Why drag topology into this article?
The new section Countable set#Topological proof of the uncountability of the real numbers seem to be out of place in this article. It is about topology, not really about countable sets. Also, from my limited knowledge of topology, the proof appears to be incorrect. Specifically, in the indiscrete topology one-point sets are not open. JRSpriggs (talk) 17:16, 13 June 2008 (UTC)
- Not disagreeing, but the indiscrete topology doesn't satisfy the hypotheses because it is non-Hausdorff. (It is in fact mentioned as a counterexample in the indicated section.) siℓℓy rabbit (talk) 17:21, 13 June 2008 (UTC)
It can make sense to include a short proof that any topological space satisfying certain simple conditions must be uncountable and perhaps that the reals do satisfy those conditions. That the reals are uncountable can be proved more simply from the fact that they are linearly ordered such that between any two points there is another and they satisfy the least-upper-bound property (you certainly DON'T need to talk about decimals, even though that's one way to do it). Michael Hardy (talk) 17:29, 13 June 2008 (UTC)
Some details missing about bijection from Q to N
How does the proof of the assertion 'Q (the set of all rational numbers) is countable' contradict the intuition that 'the cardinality of Q is larger than that of N'? Either some details are left out or it doesn't.
(0,1,0), (1,1,0), (1,1,1) and so on are not members of N. They are rather triples composed of members of N. Is there assumed another bijection from these tuples to N? If this bijection exists and is assumed, it may be useful to explicitly state it. Johanatan (talk) 23:55, 20 October 2008 (UTC)