Countability of any set with cardinality larger than that of $\mathbb N$

Question

I should preface my question by saying that I am only a college freshman finishing multivariable calculus, so please keep that in mind when answering. I would consider myself more-or-less mathematically mature, though, so don't hesitate to give me technical answers.

After reading some random things online about the countability of $\mathbb Q$, $\mathbb R$, etc., I came up with a question: how can a set that's theoretically larger than $\mathbb N$ ever be considered countable? Take $\mathbb Q$, for example. Is there anything wrong with the following "proof" that $\mathbb Q$ is not countable:

1. For every natural number $n$, we can find one unique rational number $\frac{1}{n}$. There are therefore at least $\aleph_0$ rational numbers (where $\aleph_0$ is the cardinality of $\mathbb N$).
2. Let $n = 3$. We know that there is at least one other rational number that has 3 in the denominator: 2. We thus know that there is (at least) one additional rational number $\frac{2}{3}$ besides the $\aleph_0$ we already know about. We can thus say that the cardinality of $\mathbb Q$ is at least $\aleph_0 + 1$.
3. A function is bijective iff it is both injective and surjective. A function $f : A \rightarrow B$ is injective iff every element of $A$ maps to exactly one element of $B$. A corollary of this is that for $f$ to be injective, and thus bijective, $|A|$ must not be less than $|B|$.
4. A set is countable iff there exists a bijection between the natural numbers $\mathbb N$ and the set. The cardinality of $\mathbb N$ is $\aleph_0$ and the cardinality of $\mathbb Q$ is at least $\aleph_0 + 1$. Therefore, $|\mathbb N| < |\mathbb Q|$. Because of this, there cannot be a bijection between $\mathbb N$ and $\mathbb Q$, and thus $\mathbb Q$ is not countable. QED

However, Wikipedia says that $\mathbb Q$ is countable and that its cardinality is $\aleph_0$! One possible issue may be that I butcher the concept of aleph numbers, but even if I do, doesn't the general concept still hold? How can it be that a set if countable if it is larger than $\mathbb N$?

It would be great if you can clarify any concepts I'm obviously not understanding. Thanks!

Answer 1

A set $X$ is countable if there is an surjectve map from $\mathbb{N}$ to $X$.

What you've done is shown that there is a map from $\mathbb{N}$ to $\mathbb{Q}$ which is not surjective - but this doesn't contradict the existence of some other map, from $\mathbb{N}$ to $\mathbb{Q}$, which is surjective!

Think of it this way: consider the map $f(x)=x+1$. This is a map from $\mathbb{N}$ to $\mathbb{N}$ which is not surjective; does this mean that $\mathbb{N}$ is larger than itself?

My answer to the related question Are there fewer positive integers than all integers? might be useful to you.