So the answer is no, even a magical halt oracle cannot compute the true first order theory of the standard model of arithmetic, but there are machines in which you can compute the true first order theory of the standard model of arithmetic.
Correct. Each iteration of the halting problem for oracle Turing machines (called the “Turing jump”) takes you to a new level of relative undecidability, so in particular true arithmetic is strictly harder than the halting problem.
So the answer is no, even a magical halt oracle cannot compute the true first order theory of the standard model of arithmetic, but there are machines in which you can compute the true first order theory of the standard model of arithmetic.
Correct. Each iteration of the halting problem for oracle Turing machines (called the “Turing jump”) takes you to a new level of relative undecidability, so in particular true arithmetic is strictly harder than the halting problem.