Problems With Recursively Defined Sets I am having an issue with this problem. 


  A  set, S,  of  positive  integers  is  defined  recursively  by  the  rule: 
  
  
  $1 \in S$,
  If $n\in S$, then $2n-1 \in S$
  
  
  List  all  the  elements  in  the  set $S$.


The reason I am having difficulty is because plugging in $1$ into $2n-1$ provides me with the original statement, namely, that $1\in S$

This is far as I have been able to get with the problem and I would appreciate some helpful hints.
Ok, so all that that means is that $1$ is the only element in $S$; it's the only one we put in there as a 'base', and using the 'step', it turns out we can't get any others in there.
I think that your set  $S$ is simply not well-defined...   

Taking $S_1 = \mathbb N - \{0\}$, this set satisfy the two conditions stated above.
But $S_2 = \{1\}$ too ! If such a set $S$ is well-defined via 1. &amp; 2., it must be the only set satisfying those conditions, so $S_1 = S_2 = S$... contradiction...