## General relativistic hypercomputing and foundation of mathematics

Citations: | 4 - 0 self |

### BibTeX

@MISC{Andréka_generalrelativistic,

author = {Hajnal Andréka and István Németi and Péter Németi},

title = {General relativistic hypercomputing and foundation of mathematics},

year = {}

}

### OpenURL

### Abstract

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

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Citation Context ...ch direction. It is the unifying power of logic which makes it viable to do serious work on such a diverse collection of topics. One of the main aims of the research direction represented by [1]–[3], =-=[24]-=-–[26] is to make relativity theory accessible for anyone familiar with logic. 2 Intuitive idea for non-Turing GR computing In this section we briefly recall from [31, 30] the ideas of how relativistic... |

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Citation Context ...rection. It is the unifying power of logic which makes it viable to do serious work on such a diverse collection of topics. One of the main aims of the research direction represented by [1]–[3], [24]–=-=[26]-=- is to make relativity theory accessible for anyone familiar with logic. 2 Intuitive idea for non-Turing GR computing In this section we briefly recall from [31, 30] the ideas of how relativistic comp... |

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6 |
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Citation Context ...ons concerning the geometry of rotating black holes [5], to mention only a few. The interdisciplinary character of this direction was reflected already in the 1987 course given by the present authors =-=[29]-=- during which the idea of relativistic hypercomputers emerged and which was devoted to connections between the above mentioned areas. Section 6 is also about the impact of general relativistic computi... |

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Citation Context ..., discoveries and new results have been accelerating. In the last 10 years astronomers have obtained firmer and firmer evidence for the existence of ever larger, more and more exotic black holes [39],=-=[36]-=- not to mention evidence supporting the assumption that the universe is not finite after all [41]. Nowadays the whole field is in a state of constant revolution. If the background foundation on which ... |

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5 |
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Citation Context ...of general relativistic computing as described in section 2 was found at different parts of the globe, independently. It was discovered by Németi in 1987 [29], Pitowsky in 1990 [34], Malament in 1989 =-=[27]-=-, and Hogarth in 1992 [18] independently. Németi’s idea used large slowly rotating black holes (slow Kerr spacetimes) but the careful study of feasibility and transversability of these was done later ... |

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Citation Context ...outlined train of thought can be used to show that any recursively enumerable set can be decided by a relativistic computer [15]. Actually, more than that can be done by relativistic computers. Welch =-=[44]-=- shows that the arrangement described in section 3 using Kerr black holes can solve exactly the ∆2-hard problems in the technical sense of the arithmetical hierarchy (degrees of unsolvability) used in... |

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Citation Context ...esearch direction. It is the unifying power of logic which makes it viable to do serious work on such a diverse collection of topics. One of the main aims of the research direction represented by [1]–=-=[3]-=-, [24]–[26] is to make relativity theory accessible for anyone familiar with logic. 2 Intuitive idea for non-Turing GR computing In this section we briefly recall from [31, 30] the ideas of how relati... |

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3 | Can General Relativistic Computers Break the Turing Barrier
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Citation Context ...t is not easy to ensure that this speed-up effect happens in such a way that we could utilize it for implementing some non-Turing-computable functions. In sections 2 and 3 we briefly recall from [31],=-=[30]-=- an intuitive idea of how this infinite speed-up can be achieved and how one can implement a computer based on this idea. More concrete technical details can be found in [15],[31] and to some extent i... |

3 |
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Citation Context ...esirable to re-examine the status and scope of applicability of PhCT in view of the change of our general world-picture. A relevant perspective is e.g. in Cooper [9]. Cf. also [19], [15], [31], [23], =-=[37]-=-. Assumption of an absolute time scale is a characteristic feature of the Newtonian world-view. Indeed, this absolute time has its mark on the Turing machine as a model for computer. As a contrast, in... |

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2 |
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Citation Context ...physical experiments? This leads to questions inherent in foundational issues in mathematics and physics. For more about this and about connection with Hilbert’s Programme for mathematics we refer to =-=[4]-=-. 7 Relativistic Computers and Causality Hypotheses in Physics Let us consider the hierarchy of causality hypotheses C0,...,C6 summarized in the monograph Earman [11, §6.3, pp.164-166]. None of these ... |

2 |
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Citation Context ...gard of time and space, moreover there is an infinite amount of matter-energy available in it. We will see soon that the latter infinity is not needed for our construction. For more on this see Dávid =-=[10]-=-, [31] and the references therein. Our point here is not about believing that our universe indeed has infinite time or not. The point is that assuming availability of an infinite amount of time for co... |

2 | Five views of hypercomputation
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Citation Context ...t is desirable to re-examine the status and scope of applicability of PhCT in view of the change of our general world-picture. A relevant perspective is e.g. in Cooper [9]. Cf. also [19], [15], [31], =-=[23]-=-, [37]. Assumption of an absolute time scale is a characteristic feature of the Newtonian world-view. Indeed, this absolute time has its mark on the Turing machine as a model for computer. As a contra... |

2 |
Computing the disturbance of phone lines caused by high-voltage electrical power lines
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Citation Context ...ied to matter/energy content? Is there a lower bound to mass which is needed to store 10 bits of information? This is a question which has nagged one of the authors ever since he wrote his MsC thesis =-=[28]-=- where a separate section was devoted to this issue. The question is: “If I want to write more, do I need more paper to write on?” Right now it seems to us that the answer is in the negative. Matter a... |

1 |
Supertasks: Gödel strikes back
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(Show Context)
Citation Context ...tsburgh (Earman et al). The first thorough and systematic study of relativistic computation was probably Hogarth [19]. Related work on relativistic computing include [44], [37], [11, §4], [12], [13], =-=[17]-=-, [30]. 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,... |