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.

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

Assessment comment

The comment(s) below were originally left at Talk:Paradoxical set/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

The example with the $\mathbb {N}$ is not good. The function $f$ specified is not one-to-one and hence not invertible, and so it is not clear what 'the group of functions $G$ generated by the natural function $f$' means. (Also, a little quibble, the specified function doesn't map the odd numbers onto $\mathbb {N}$ in the more frequent definition of $\mathbb {N}$ that contains zero.) Perhaps one should say that the 'universe set' is the real numbers $\mathbb {R}$, and the group $G$ is the group of bijections of $\mathbb {R}$ generated by the two invertible functions $f_{1}(x)=x/2$ and $f_{2}(x)=(x-1)/2$ .

Last edited at 00:51, 7 April 2009 (UTC). Substituted at 02:26, 5 May 2016 (UTC)