TITLE: How many subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$?
QUESTION [14 upvotes]: How many subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$?

I realize that any of the class of functions $f:x\to (n\cdot x)$ gives a bijection between $\mathbb{N}$ and the subset of $\mathbb{N}$ whose members equal multiples of n.  So, we have at least a countable infinity of sets which have the same cardinality of $\mathbb{N}$.  But, we could remove a single element from any countably infinity subset of the natural numbers and we still will end up with a countably infinite subset of $\mathbb{N}$.  So (the reasoning here doesn't seem quite right to me), do there exist uncountably many infinite subsets of $\mathbb{N}$ with the same cardinality of $\mathbb{N}$?  
Also, is the class of all bijections $f: \mathbb{N} \to \mathbb{N}$ and a given countably infinite subset uncountably infinite or countably infinite?

REPLY [2 votes]: Here is a simple explicit bijection from $\mathcal P(\mathbb N)$ to $\mathcal P(\mathbb N)\setminus \operatorname{Fin}(\mathbb N)$:
$$ f(A) = \begin{cases} \{ n+1 \mid n\in A \} & \text{if $A$ is cofinite} \\
\mathbb N \setminus \{ n+1 \mid n\in A \} & \text{if $A$ is finite} \\
A & \text{if $A$ is neither finite nor cofinite} \end{cases} $$
($A$ is said to be cofinite iff $\mathbb N\setminus A$ is finite).