Completeness and consistency of system of calculus

It is widely held that ZFC cannot be shown to be self-consistent or complete.

So, what happens to the system of calculus? Can it be shown to be complete or self-consistent? (Edit: Oops. So, two questions here: Can the system itself prove its own consistency and completeness? And what system of logic can prove its consistency?)

By calculus, I mean univariable/multivariable integration, differentiation and limit.

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Widely held? Except a few cranks I'd think it's more than "widely held" that ZFC cannot prove its own consistency, unless it is inconsistent. –  Asaf Karagila Feb 2 '13 at 3:31
@AsafKaragila true.. but just to be conservative :) –  user60629 Feb 2 '13 at 3:35

The portion of calculus you mention is mainly relying on the model $\mathbb R$ or real numbers. Very rigorous models of the real numbers can be constructed from models of the rationals, which in turn can be constructed from models of the integers, which in turn can be constructed from models of the naturals, which in turn can be constructed from any model of $ZF$. The latter can't be proved consistent (unless it is inconsistent).