Turing machines can’t use third order logic. That might be a limitation of the programmer, or it might be a limitation of deterministic machines. A Turing machine can’t ever compute anything about it’s own output; third order logic might allow deterministic machines which can compute the results of their own output.
I suspect that there is a halting problem where the halting status of machines which only use second order logic cannot be computed in third order logic, just like limits cannot be computed in Peano arithmetic.
Turing machines can’t use third order logic. That might be a limitation of the programmer, or it might be a limitation of deterministic machines. A Turing machine can’t ever compute anything about it’s own output; third order logic might allow deterministic machines which can compute the results of their own output.
I suspect that there is a halting problem where the halting status of machines which only use second order logic cannot be computed in third order logic, just like limits cannot be computed in Peano arithmetic.