Very true, but that kills the notion that objectivism is the only "logical" system of philosophy. If you accept the premises as axioms, then you can construct the objectivist system in a perfectly logical manner, but since there's no way to prove the axioms, there's no real reason to accept the axioms of objectivism over some other set of axioms. And if I then choose to substitute some other set of unprovable axioms - such as "God exists and he says X is wrong" - then I can construct some completely different system in a perfectly logical manner, just as logically as I can construct the objectivist philosophy.
No philosophy is completely self-contained and completely provable - it can't be, as you quite rightly point out. That doesn't make objectivism inherently inferior to other philosophies, but it ends any nonsense about it being inherently superior.
Inferiority and superiority depend on the value of the axioms underlying them. A good axiom, one that will lead to a superior philosophy, is one that is so obvious people stare at you as if to say, "Well, of course that's true, why would you even mention it." When you get axioms not rooted in cultures, you're getting somewhere.
Can you get to one philosophy that is superior to all others? It will have to be pretty generic. The only axiom I can think of is it must be based on liberty.