Beyond Undecidability

Abstract

We know that virtually all interesting questions about Turing machines—whether a given TM halts on a given input, whether a given TM accepts a finite set, and so on—are undecidable. But are all these questions equally hard? For example, suppose by some magic we were given the power to decide the halting problem. Could we somehow use that power to decide if a given TM accepts a finite set? In other words, relative to the halting problem, is finiteness decidable?