TITLE: If the set of odd numbers is a subset of $\mathbb N$ then surely it is smaller than $\mathbb N$
QUESTION [14 upvotes]: 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 $\mathbb{O}$?

REPLY [2 votes]: A set is called "Dedekind infinite" if and only if it can be put into one-to-one correspondence with a proper subset of itself.
Example: N = {1, 2, 3, ...} = set of natural numbers is in 1-1 correspondence with O = {odds}, using f(n) = 2n-1, as already noted.
One can prove that a set is Dedekind infinite if and only if it is infinite (in usual sense). It is for infinite sets that we have the seeming paradox noted so far about subsets.
The answer is that we don't use terms like "larger" based on the idea of the subset relation. It is simply not done because it is not useful and it is even confusing insofar as it overlooks this seeming paradox.
N and O have the same cardinality, owing to the bijection f(n) = 2n-1. The fact that O is a proper subset of N is initially confusing, but we never speak of N being "larger than" O.
BONUS: This reminds me of a puzzle. You have infinitely many rooms numbered 1, 2, 3, ... ad infinitum, each filled with a rock. Each room has exactly one rock, and there are no other rooms. I come to you and bring one more rock. How can I arrange to have each and every rock (mine plus originals) in a room, but with no two rocks in one room?
VARIATION: Same as before, but I bring my own infinity of rocks, numbered 1, 2, 3, ... and want to combine with existing infinity so as to have each rock in a room, with no two rocks sharing a room. How?
Have fun!
(Dr. Michael W. Ecker is a retired Penn State U. associate professor of mathematics. He also was editor-publisher of Recreational & Educational Computing in addition to holding other positions.)