Defining a set of prime numbers up to index $m$.

Question

I want to define a set of prime numbers up to index $m$ as follows $$P_m=\{p_i:p_1\leq p_i\leq p_m,i\in \mathbb N\}$$ Now, $m=1,2,3,...$ should define the cardinality of this set. I like this compactness but am concerned that the $i\in\mathbb N$ implies that indices across all natural numbers must be chosen rather than stopping at index $m$. I originally did this for more generality such that $m$ can go to infinity. Should I instead define a subset of $\mathbb N$?