If the set of odd numbers is a subset of $\mathbb N$ then surely it is smaller than $\mathbb N$ I have been studying Cantor's theorem, and I follow entirely that the set of natural numbers $\mathbb{N}$ is countable, as is the set of odd numbers (let's call it $\mathbb{O}$).
I understand his proof that there is a correspondence between the two sets and feel like I could accept that they therefore have the same cardinality based on that ... but ... hold on ...
We know that $\mathbb{O}$ is a subset of $\mathbb{N}$ right? $\mathbb{N}$ certainly contains all of the odd numbers. 
We also know that $\mathbb{N}$ contains numbers that are not odd. 
So if $\mathbb{N}$ contains all of $\mathbb{O}$ and some other stuff as well then surely it would be totally justified to argue that $\mathbb{N}$ is larger than $\mathbb{O}$?

Consider the correspondence
$n \to 4n-1$
I.e.,
$1\to 3$
$2\to 7$
Etc.
Now it looks like there only enough natural numbers to cover a proper subset of the odds.  So does that mean there are more odd natural numbers than natural numbers?
