If you try to use mathematical induction to form a theory that includes all such statements, that theory will have an infinite number of postulates and will not be able to be analyzed by a Turing machine.

This part is not quite accurate. Actually, the commonly used theories of arithmetic (and sets) have infinitely many axioms. The actually problem with your approach above is that the theory still won't be able to prove its own consistency since any proof can only use finitely many of the axioms. One can of course add an additional axiom and keep going using transfinite induction, but now one will finally run into a theory that a Turing machine can't analyze.