What is the decision version of independent set?
The definition of the decision version of independent set is the following:

Given as input a graph $G = (V,E)$ and an integer $k \in \mathbb{N}$, does there exist a set of pairwise non-adjacent vertices $I \subseteq V$ of size $|I| \geq k$?

The problem is polynomial time solvable if you consider $k$ to be constant, i.e., you can solve the problem in time $\mathcal{O}(n^k)$ but we usually take $k$ to be a part of the input.  If you reduce an NP-complete problem to this problem, you will see that the $k$ is indeed unbounded.