On the Possibility, or Otherwise, of Hypercomputation
PHILIP D. WELCH · The British Journal for the Philosophy of Science · 2004
We claim that a recent article of P. Cotogno ([2003]) in this journal is based on an incorrect argument concerning the non-computability of diagonal functions. The point is that whilst diagonal functions are not computable by any function of the class over which they diagonalise, there is no ‘logical incomputability’ in their being computed over a wider class. Hence this ‘logical incomputability’ regrettably cannot be used in his argument that no hypercomputation can compute the Halting problem. This seems to lead him into a further error in his analysis of the supposed conventional status of the infinite time Turing machines of Hamkins and Lewis ([2000]). Theorem 1 refutes this directly. 1. The diagonalisation misunderstanding 2. Infinite computation 3. Conclusion