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 $\mathbb{O}$?
It depends on your definition of "larger". Once you make that precise, you'll answer your own question.
Imagine that you didn't know that $\mathbb O$ is a subset of $\mathbb N$, or that they have any elements in common. Imagine that you couldn't even inspect the properties of the elements of $\mathbb O$ and $\mathbb N$. Imagine that you didn't even recognize those elements as numbers. To paint a picture, imagine that someone takes a copy of $\mathbb O$ and a copy of $\mathbb N$ and encloses all of their elements in whimsically colored hardwood boxes.
Now, all you know is that $\mathbb N=\{a,b,c,d,\ldots\}$ and $\mathbb O=\{A,B,C,D,\ldots\}$, where $a$ denotes the teal-striped rosewood box, etc. Maybe $a$ contains the number $0$. Maybe $B$ contains the number $3$. But you have no way of knowing that.
Under these restrictions, how will you compare the sizes of $\mathbb O$ and $\mathbb N$? Can you come up with any justification that one is larger than the other?
For the following lists,
$$1,2,3,...$$
$$1,3,5,...$$
erase the labels, and replace them by dots:
$$\small{\bullet,\;\bullet,\;\bullet,\;...}$$
$$\small{\bullet,\;\bullet,\;\bullet,\;...}$$
Now which list has more elements?
The key idea is that for comparing the "sizes" of two sets, labels shouldn't matter. What matters for determining equal size is one-to-one correspondence.