Partitioning an infinite set Can you partition an infinite set, into an infinite number of infinite sets?

An (imo) beautiful practical example from Elementary Number Theory.
As you know, any equivalence relation, partitions a set.
Now, if you partition the Natural numbers $(\mathbb{N})$ = {1, 2, 3, ... } ( 0 of course NOT included ), based on the relation "Prime Signature", you have what you are looking for.
Based on the Main Theorem of Arithmetic we can map any natural number to two ordered lists:
$$
n = \Pi_{i=1}^{\omega(n)} p_i^{\alpha^{i}}.
$$
For example: $12 = 2^2 3^1$, 30= $2^1 3^1 5^1$, and so on, where $12$ belongs to the partition $\{2,1\}$ and $30$ belongs to the partition $\{1,1,1\}$.
We call the ordered list of ${\alpha_{i}}'s$ the "Prime Signature". ( Example: $1000$ has signature $\{3,3\}$ ).
What makes this infinite partition so 'beautiful' is that every subset of $\mathbb{N}$ in this infinite partition corresponds to a partition of a natural number.
