More the number of for loops greater the problems solved
Let us consider the LOOP programming language. For a program $f$, denote by $f_{\max}(n)$ the maximal value of a variable at the end of the program, given that initially, all variables are at most $n$. If $f$ has only one loop then $f_{\max}(n) = O(n)$. In contrast, using two nested loops you can compute the product function, and the resulting program $g$ satisfies $g_{\max} = \Theta(n^2)$. This separates $P_1$ and $P_2$.

More generally, $f_{\max}$ can be bounded in terms of the maximum nesting depth of loops in $f$. If the maximum nesting depth is $d$, then the growing rate of $f_{\max}$ is comparable to the $d$th level of the Ackermann hierarchy. This allows us to separate $P_i$ and $P_{i+1}$ for every $i$.