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 $\mathbb{O}$?
Best Answer
It depends on your definition of "larger". Once you make that precise, you'll answer your own question.