[PHI 3800] Can science explain everything?

Is the question "Can science explain everything?" a sensible (or even a coherent) question?

Can everything be explained (whether scientifically or not)?

Here's why one might think that the answer to the latter question is "no."

1. Nothing can explain itself. [explanandum ≠ explanans]
2. If X explains Y, then one could ask what explains X.
3. If Z explains X, then one could ask what explains Z.
4. ...and so on, ad infinitum.
5. The chain of explanation is either infinite or finite.
6. If the chain of explanation is infinite, then there must be something that remains unexplained (since we cannot provide explanations ad infinitum).
7. If the chain of explanation is finite, then it stops at some arbitrary point, thus leaving every subsequent point in the chain of explanation dependent on a point that cannot explain its successor.
8. (Therefore) Either way, there must be something that remains unexplained.

What do you make of this argument?