Abstract. We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Every Π1 1 set, for example, is decidable by such machines, and the semi-decidable sets form a portion of the ∆1 2 sets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators. In these days of super-fast computers whose speed seems to be increasing without bound, the more philosophical among us are perhaps pushed to wonder: what could we compute with an infinitely fast computer? By proposing a natural model for supertasks—computations with infinitely many steps—we provide in this paper a theoretical foundation on which to answer this question. Our model is simple: we simply extend the Turing machine concept into transfinite ordinal time. The resulting machines can perform infinitely many steps of computation, and go on to more computation after that. But mechanically they work just like Turing machines. In particular, they have the usual Turing machine hardware; there is still the same smooth infinite paper tape and the same mechanical head moving back and forth according to a finite algorithm, with finitely many states. What is new is the definition of the behavior of the machine at limit ordinal times. The resulting computability theory leads to a notion of computation on the reals, concepts of decidability and semi-decidability for sets of reals as well as individual reals, two kinds of jump-operator, and a notion of relative computability using oracles which gives a rich degree structure on both the collection of reals and the collection of sets of reals. But much remains unknown; we hope to stir interest in these ideas, which have been a joy for us to think about.

We examine the current status of the physical version of the Church-Turing Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with Deutsch et al [17] that PhCT is not only a conjecture of mathematics but rather a conjecture of a combination of theoretical physics, mathematics and, in some sense, cosmology. Since the idea of computability is intimately connected with the nature of Time, relevance of spacetime theory seems to be unquestionable. We will see that recent developments in spacetime theory show that temporal developments may exhibit features that traditionally seemed impossible or absurd. We will see that recent results point in the direction that the possibility of artificial systems computing non-Turing computable functions may be consistent with spacetime theory. All these trigger new open questions and new research directions for spacetime theory, cosmology, and computability.

In the literature on the compatibility between the time of our experience and the time of physics, the special theory of relativity has enjoyed central stage. By bringing into the discussion the general theory of relativity, I suggest a new analysis of the misunderstood notion of becoming, developed from hints in Gödel’s published and unpublished arguments for the ideality of time. I claim that recent endorsements of such arguments, based on Gödel’s own “rotating ” solution to Einstein’s field equation, fail: once understood in the right way, becoming can be shown to be both mind-independent and compatible with spacetime physics. Being a needed tertium quid between views of time traditionally regarded as in conflict, such a new approach to becoming should also help to dissolve a crucial aspect of the century-old debate between the so-called A and B theories of time.

The principle of common cause is discussed as a possible fundamental principle of physics. Some revisions of Reichenbach’s formulation of the principle are given, which lead to a version given by Bell. Various similar forms are compared and some equivalence results proved. The further problems of causality in a quantal system, and indeterministic causal structure, are addressed, with a view to defining a causality principle applicable to quantum gravity. 1

2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4

Abstract.- Can general relativistic computers break the Turing barrier?- Are there final limits to human knowledge?- Limitative results versus human creativity (paradigm shifts).- Gödel’s logical results in comparison/combination with Gödel’s relativistic results.- Can Hilbert’s programme be carried through after all? 1 Aims, perspective The Physical Church-Turing Thesis, PhCT, is the conjecture that whatever physical computing device (in the broader sense) or physical thought experiment will be designed by any future civilization, it will always be simulatable by a Turing machine. The PhCT was formulated and generally accepted in the 1930’s. At that time a general consensus was reached declaring PhCT valid, and indeed in the succeeding decades the PhCT was an extremely useful and valuable maxim in elaborating the foundations of theoretical computer science, logic, foundation of mathematics and related areas. But since PhCT is partly a physical conjecture, we emphasize that this consensus of the 1930’s was based on the physical worldview of the 1930’s. Moreover, many thinkers considered PhCT as being based on

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

impact on the physics of the

Abstract. This paper consists mostly of pictures visualizing ideas leading to Gödel’s rotating cosmological model. The pictures are constructed according to concrete metric tensor fields. The main aim is to visualize ideas. Some kinds of physical theories describe what our universe looks like. Other kinds of physical theories describe instead what the universe could be like independently of the properties of the actual universe. This second kind aims for the “basic laws of physics ” in some sense which we will not make precise here (but cf. e.g. Malament [Mal84, pp.98–99]). The present paper belongs to the second kind. Moreover, it is even more abstract than this, namely it aims for visualizing or grasping some mathematical or logical aspects of what the universe could be like. The first few pages of this material are of a “science-popularizing ” character in the sense that first we recall a space-time diagram from Hawking–Ellis [HE73] as “God-given truth”, i.e. we do not explain why the reader should believe that diagram. Then we derive in an easily understandable visual manner an exciting, exotic consequence of that diagram: time-travel. This applies to the first few pages.

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