Problems With Recursively Defined Sets

Question

I am having an issue with this problem.

A set, S, of positive integers is defined recursively by the rule:

1. $1 \in S$,

2. 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.

Answer 1

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.

Answer 2

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. & 2., it must be the only set satisfying those conditions, so $S_1 = S_2 = S$... contradiction...