K-theory, proper class, set, isomorphism types

Question

Define $K$ as the free abelian group with generators $[A],[A'],[A''],\dotsc$, the equivalence classes of isomorphism types, modulo $[A]=[A']+[A'']$ where $0\to A'\to A \to A''\to0$ is a s.e.s. of modules in $R\mathsf{-Mod}$. Is $K$ a set or a proper class? What is all the $A$'s are from $R\mathsf{-mod}$ (the f.g. left $R$-modules); again is $K$ a set or a proper class? Why?