The true first-order theory of the standard model of arithmetic has Turing degree 0ω. That is to say, with an oracle for true arithmetic, you could decide the halting problem, but also the halting problem for oracle Turing machines with a halting-problem-for-regular-Turing-machines oracle, and the halting problem for oracle Turing machines with a halting oracle for those oracle Turing machines, and so on for any finite number of iterations. Conversely, if you had an oracle that solves the halting problem for any of these finitely-iterated-halting-problem-oracle Turing machines, you could decide true arithmetic.
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.
The true first-order theory of the standard model of arithmetic has Turing degree 0ω. That is to say, with an oracle for true arithmetic, you could decide the halting problem, but also the halting problem for oracle Turing machines with a halting-problem-for-regular-Turing-machines oracle, and the halting problem for oracle Turing machines with a halting oracle for those oracle Turing machines, and so on for any finite number of iterations. Conversely, if you had an oracle that solves the halting problem for any of these finitely-iterated-halting-problem-oracle Turing machines, you could decide true arithmetic.
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.