On the Physical Possibility of Ordinal Computation

α-recursion lifts classical recursion theory from the first transfinite ordinal ω to an arbitrary admissible ordinal α [13]. Turing machine models for α-recursion and other types of transfinite computation have been proposed and studied [5] and [7] and are applicable in computational approaches to the foundations of logic and mathematics [11]. They also provide a natural setting for modeling extensions of the algorithmic logic described in [2] and [3]. Such α-Turing machines can complete a θ-step computation for any ordinal θ < α. Here we consider constraints on the physical realization of α-Turing machines that arise from the structure of physical spacetime. In particular, we show that an α-Turing machine is realizable in a spacetime constructed from R only if α is countable. Further, while there are spacetimes where uncountable computations are possible, there is good reason to suppose that such nonstandard spacetimes are nonphysical. We conclude with a suggestion for a revision of Church’s thesis appropriate as an upper bound for physical computation. 1. Ordinal Recursion and Physical Computation Church’s thesis is that every computable function is recursive. Since Turing machines provide a computational model for the recursive functions, a function is Church computable only if it is Turing computable. But insofar as one takes what is actually computable to be a question of fact, one cannot simply stipulate the answer. More specifically, one should expect that what is in fact computable ultimately depends on the nature of the physical world. On this view, questions of what is computable must be answered relative to (i) a specified physical theory (and other assorted background physical assumptions) and (ii) a computational model described relative to these physical assumptions. Many of our best physical theories allow that what is physically computable may extend well beyond what is Turing computable. But how much more than Turing computable might be physically possible? And for that matter, how should one measure the strength of computations that extend beyond the Turing computable functions? α-recursion theory provides one context for answering such questions. α-recursion lifts classical recursion theory from the first transfinite ordinal ω to an arbitrary admissible ordinal α [13]. If one thinks of Turing machines as providing natural computational models for ω-recursive functions, then one might wish to have similar computational models for α-recursive functions. Following earlier descriptions of infinite time Turing machines [5], Peter Koepke describes an Date: October 23, 2009. 1Here we will assume a version of classical mechanics with no energy or velocity constraints and an α-recursive Turing model of computation. More specific auxiliary assumptions will be described as needed.