Talk:Paradoxical set

Natural numbers example
For the example given for the natural numbers, the given function, as of may 26, 2009, (n maps to n/2 or (n-1)/2 depending on parity) is not injective, so it is not clear how it generates a group. —Preceding unsigned comment added by 204.60.65.162 (talk) 03:31, 27 May 2009 (UTC)

Therefore I removed the following section from the article here, sice it is a chat, rather than encyclopedic text. And unreferenced, too. Max Longint (talk) 00:53, 17 December 2011 (UTC)

Natural numbers
An example of a paradoxical set is the natural numbers. They are paradoxical with respect to the group of functions $$G$$ generated by the natural function $$f$$:

$$ f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ (n+1)/2, & \mbox{if }n\mbox{ is odd} \end{cases} $$

Split the natural numbers into the odds and the evens. The function $$f$$ maps both sets onto the whole of $$\mathbb{N}$$. Since only finitely many functions were needed, the naturals are $$G$$-paradoxical. This example does not work: the natural function is not injective, and as such has no well defined inverse. Therefore it does not generate a group of functions.

So, what's the deal here? Max Longint (talk) 00:53, 17 December 2011 (UTC)

Assessment comment
Substituted at 02:26, 5 May 2016 (UTC)