Is there a non-contradictory non-trivial axiomatic system in which Gödel's theorem is undecidable?

Question

Gödel's Incompleteness Theorem states that in any non-contradictory non-trivial axiomatic system there are certain statements or theorems whom cannot be proved in that system,

For example in ZFC, assuming it is non-contradictory, there is the famed Continuum Hypothesis and Whitehead's problem. My question is, is there any such system (but different from ZFC) in which Gödel's Incompleteness Theorem itself is an unprovable statement?

Answer 1

Robinson arithmetic is non-trivial enough that the incompleteness theorem applies to it, but as far as I know not strong enough to prove the incompleteness theorem itself.

Answer 2

Just a footnote to Henning Makholm's answer. You indeed need to look at very weak systems for examples of the phenomenon you want. For a start, as soon as you add a little bit of induction to Robinson Arithmetic (officially, induction for $\Sigma_1$ wffs) you get a theory that, for any recursively axiomatized $T$ which includes a bit of arithmetic and any sensible system of Gödel-coding, can show that $Con_T \to \neg Prov_T(\overline{\ulcorner G_T\urcorner)}$, i.e. it can prove the formalized version of (half of) Gödel's incompleteness theorem for $T$. (It was rather important to Gödel that the incompleteness theorem is elementary in this sort of way.)