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General relativistic hypercomputing and foundation of mathematics

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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.

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Notes

  1. As a contrast, one of the founding fathers of PhCT, László Kalmár, always hoped for a refutation of PhCT and to his students he emphasized that PhCT is meant to be a challenge to future generations, it is aimed at “teasing” researchers to put efforts into attacking PhCT (Kalmár 1959).

  2. Measurable e.g. by radar.

  3. This is a technical expression in observational astronomy.

  4. Very tentatively: Recently, emerging new kinds of computing devices like the Internet seem to pose a challenge against the conjecture in Question 1 (cf. Wiedermann and van Leeuwen 2002). However, even the Internet (or even the Human Mind) will probably not prove ZFC consistent (assuming C6). So, perhaps we should separate PhCT into two theses, one about “hard” problems like proving the consistency of ZFC, and the other about problems in general which are not Turing computable.

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Acknowledgements

Special thanks are due to Gyula Dávid for intensive cooperation and ideas on this subject. We are grateful for enjoyable discussions to Attila Andai, John Earman, Gábor Etesi, Stefan Gruner, Balázs Gyenis, Mark Hogarth, Judit Madarász, Victor Pambuccian, Gábor Sági, Endre Szabó, László E. Szabó, Renáta Tordai, Sándor Vályi, Jir̂i Wiedermann, Chris Wüthrich. We gratefully acknowledge support by the Hungarian National Foundation for scientific research grant No. T73601.

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  1. Rényi Institute of Mathematics, P.O. Box 127, 1364, Budapest, Hungary

    Hajnal Andréka, István Németi & Péter Németi

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  1. Hajnal Andréka
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Correspondence to Hajnal Andréka.

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Andréka, H., Németi, I. & Németi, P. General relativistic hypercomputing and foundation of mathematics. Nat Comput 8, 499–516 (2009). https://doi.org/10.1007/s11047-009-9114-3

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