The axiom scheme of replacement is very natural, while also avoidable in most "mainstream" mathematical practices; I know there are execptions such as Borel determinacy. My question is whether ZC (ZFC without replacement) is still an interesting set theory? For example the Continuum Hypothesis seems to make sense: whether there exists a bijection between $\mathcal{P}(\mathbb{N})$ and the set of isomorphism classes of well orderings on $\mathbb{N}$. I guess we actually can construct all $\aleph_n$, though not in the form of von-Neumann ordinal.
The above question may be too naive, since a statement independent of ZFC is of course independent of ZC. So here are some more precise questions:
Is establishing the independence of CH any easier if the base theory is ZC?
Most of ordinary mathematics is about sets in $V_{\omega+\omega}$. How much can the theory of $V_{\omega+\omega}$ be changed using current set theory techniques?