A Brief Critique of Pure Hypercomputation
Hypercomputationthe hypothesis that Turing-incomputable objects can be computed through infinitary meansis ineffective, as the unsolvability of the halting problem for Turing machines depends just on the absence of a definite value for some paradoxical construction; nature and quantity of computing resources are immaterial. The assumption that the halting problem is solved by oracles of higher Turing degree amounts just to postulation; infinite-time oracles are not actually solving paradoxes, but simply assigning them conventional values. Special values for non-terminating processes are likewise irrelevant, since diagonalization can cover any amount of value assignments. This should not be construed as a restriction of computing power: Turings uncomputability is not a barrier to be broken, but simply an effect of the expressive power of consistent programming systems.
hypercomputation halting problem diagonalization super-task zeno machine