In complexity theory, we do not call a decision problem that is not in NP "NP-complete".
But in computability, do we call a machine model "Turing complete" if it can compute functions which Turing machines can not?
The definition from Wikipedia didn't address this problem explicitly, and likely assumed yes:
A computational system that can compute every Turing-computable function is called Turing complete (or Turing powerful). Alternatively, such a system is one that can simulate a universal Turing machine.
For example, if there is a system that can:
- Compute every Turing-computable function.
- Compute the halting problem for a Turing machine.
And nothing else. Is it Turing complete?