A version of the Church-Turing Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logico-mathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, non-well-founded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard Church-Turing Thesis.

Abstract. Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a non-Turing computable task, e.g. which can decide the halting problem of Turing machines or decide whether ZF set theory is consistent (more precisely, can decide the theorems of ZF). Starting from this, we will discuss the impact of recent breakthrough results of relativity theory, black hole physics and cosmology to well established foundational issues of computability theory as well as to logic. We find that the unexpected, revolutionary results in the mentioned branches of science force us to reconsider the status of the physical Church Thesis and to consider it as being seriously challenged. We will outline the consequences of all this for the foundation of mathematics (e.g. to Hilbert’s programme). Observational, empirical evidence will be quoted to show that the statements above do not require any assumption of some physical universe outside of our own one: in our specific physical universe there seem to exist regions of spacetime supporting potential non-Turing computations. Additionally, new “engineering ” ideas will be outlined for solving the so-called blue-shift problem of GR-computing. Connections with related talks at the Physics and Computation meeting, e.g. those of Jerome Durand-Lose, Mark Hogarth and Martin Ziegler, will be indicated. 1

Abstract. We discuss the impact of very recent developments of spacetime theory, black hole physics, and cosmology to well established foundational issues of computability theory and logic. Namely, we describe a physical device in relativistic spacetime which can compute a non-Turing computable task, e.g. which can decide the halting problem of Turing machines or whether ZF set theory is consistent or not. Connections with foundation of mathematics and foundation of spacetime theory will be discussed. 1

This paper investigates the view that digital hypercomputing is a good reason for rejection or re-interpretation of the Church-Turing thesis. After suggestion that such re-interpretation is historically problematic and often involves attack on a straw man (the ‘maximality thesis’), it discusses proposals for digital hypercomputing with “Zeno-machines”, i.e. computing machines that compute an infinite number of computing steps in finite time, thus performing supertasks. It argues that effective computing with Zeno-machines falls into a dilemma: either they are specified such that they do not have output states, or they are specified such that they do have output states, but involve contradiction. Repairs though noneffective methods or special rules for semi-decidable problems are sought, but not found. The paper concludes that hypercomputing supertasks are impossible in the actual world and thus no reason for rejection of the Church-Turing thesis in its traditional interpretation. 1