## First-order logic foundation of relativity theories (2006)

Venue: | In New Logics for the XXIst Century II, Mathematical Problems from Applied Logics, volume 5 of International Mathematical Series |

Citations: | 8 - 8 self |

### BibTeX

@INPROCEEDINGS{Madarász06first-orderlogic,

author = {Judit X. Madarász and István Németi and Gergely Székely},

title = {First-order logic foundation of relativity theories},

booktitle = {In New Logics for the XXIst Century II, Mathematical Problems from Applied Logics, volume 5 of International Mathematical Series},

year = {2006},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity— including such revolutionary areas as black hole physics, relativistic computers, new cosmology—are presented in this paper. We would like to invite the logician reader to take part in this grand enterprise of the new century. Besides general perspective and motivation, we present initial results in this direction.

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Citation Context ...Curves (CTC). Such are Gödel’s rotating universe [25], Kerr and Kerr – Newman spacetimes [47], Gott’s spacetime [27], Tipler’s rotating cylinder [64], van Stockum’s spacetime [57], Taub-NUT spacetime =-=[31]-=-, to mention only a few. Since timelike curves correspond to possible life-lines of observers, in these spacetimes an observer can go through the same event more than once. This situation can be inter... |

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Citation Context ...ng barrier. There are several models of GR in which there are so-called Closed Timelike Curves (CTC). Such are Gödel’s rotating universe [25], Kerr and Kerr – Newman spacetimes [47], Gott’s spacetime =-=[27]-=-, Tipler’s rotating cylinder [64], van Stockum’s spacetime [57], Taub-NUT spacetime [31], to mention only a few. Since timelike curves correspond to possible life-lines of observers, in these spacetim... |

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Citation Context ...ts occur are called Malament – Hogarth spacetimes, cf. e.g., Earman [15, §4], [38]. In Malament – Hogarth spacetimes, we can design a computer that decides non-Turing computable sets, cf. e.g., [38], =-=[16]-=-, [14], [19]. Thus inside these spacetimes, we can decide whether an axiom system of set theory (for example ZFC) is consistent or not. Therefore, in contrast with the consequenceFOL FOUNDATION OF RE... |

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Citation Context ...st interest of logic community to apply logic to spacetime theory, relativity, cosmology, and black hole physics. Indeed, logic can benefit from such studies in many ways. As a bonus, as indicated in =-=[14]-=- or [38], spacetime theory can give a feedback to the foundation of mathematics itself. For certain reasons, the foundation of mathematics has been carried through strictly within the framework of fir... |

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Citation Context ...els of GR in which there are so-called Closed Timelike Curves (CTC). Such are Gödel’s rotating universe [25], Kerr and Kerr – Newman spacetimes [47], Gott’s spacetime [27], Tipler’s rotating cylinder =-=[64]-=-, van Stockum’s spacetime [57], Taub-NUT spacetime [31], to mention only a few. Since timelike curves correspond to possible life-lines of observers, in these spacetimes an observer can go through the... |

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Citation Context ...res of the above mentioned three spacetimes coincide. Is this a coincidence or is there a more general “theory of rotating spacetimes” lurking in the background. For more on this question we refer to =-=[5]-=-. In particular, we are looking for a logical answer to the quasi-philosophical question: “Exactly why and how CTC’s are generated in rotating black holes and in Gödel’s universe. Why do they counter-... |

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Citation Context ...uters breaking the Turing barrier. There are several models of GR in which there are so-called Closed Timelike Curves (CTC). Such are Gödel’s rotating universe [25], Kerr and Kerr – Newman spacetimes =-=[47]-=-, Gott’s spacetime [27], Tipler’s rotating cylinder [64], van Stockum’s spacetime [57], Taub-NUT spacetime [31], to mention only a few. Since timelike curves correspond to possible life-lines of obser... |

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Citation Context ...eness theorem, see for example, Väänänen [65, p.505]. For more motivation for staying inside FOL as opposed to higher-order logic, see for example, [1], [2, Appendix 1: “Why exactly FOL”], [6], [21], =-=[48]-=-, [67]. The same reasons motivate the effort of keeping the foundation of spacetime and relativity theory inside FOL. The interplay between logic and relativity theory goes back to around 1920 and has... |

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Citation Context ...Kerr – Newman spacetimes that are associated with charged, rotating and charged-rotating black holes, respectively. For astronomical evidence for the existence of rotating black holes cf. e.g., [49], =-=[58]-=-. In these spacetimes, there is an event whose causal past contains timelike curves which are infinitely long in the future direction. Such a curve can be the life-line of an observer (or computer) wh... |

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Citation Context ...tions, [55]. So logic seems to be best suited for providing rational understanding of situations like the circularity represented by CTC’s or time travel. For more on CTC’s, cf. e.g., [15, §6], [17], =-=[28]-=-. These are only a few of the many examples that show that turning Relativity Theory into a real FOL theory, axiomatizing it and analyzing its logical structure seem to be a promising, worthwile under... |

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Citation Context ...e next axiom. AxEOF: A FOL axiom stating that the quantity part 〈Q; +, ·, ≤〉 is a Euclidean 1 ordered field. For the first-order definition of linearly ordered field, see for example, Chang – Keisler =-=[11]-=-. We will need some definitions to formulate our other axioms. Let 0, 1, −, /, √ be the usual field operations which are definable from “+” and “·”. We use the vector-space structure of Q d , i.e. if ... |

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Citation Context ...thematics was initiated by several authors including David Hilbert [34], cf. also Hilbert’s 6th problem [33], Patrick Suppes [59], Alfred Tarski [32] and leading contemporary logician Harvey Friedman =-=[22, 23]-=-. There are several reasons for seeking an axiomatic foundation of a physical theory [60]. One is that the theory may be better understood by providing a basis of explicit postulates for the theory. A... |

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Citation Context ...est of logic community to apply logic to spacetime theory, relativity, cosmology, and black hole physics. Indeed, logic can benefit from such studies in many ways. As a bonus, as indicated in [14] or =-=[38]-=-, spacetime theory can give a feedback to the foundation of mathematics itself. For certain reasons, the foundation of mathematics has been carried through strictly within the framework of first-order... |

7 | Twin Paradox and the Logical Foundation of Relativity Theory
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Citation Context ...rtial observers and for SR can be found in the literature ([6], [26], [1]), axiom systems— let alone FOL axiom systems—for accelerated observers and for GR are not too many in the literature (but cf. =-=[44]-=- for an exception). In section 4, we recall a streamlined FOL axiomatization AccRel of SR extended with accelerated observers. In section 5, we take one step toward GR and investigate an aspect of tim... |

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Citation Context ...ompleteness theorem, see for example, Väänänen [65, p.505]. For more motivation for staying inside FOL as opposed to higher-order logic, see for example, [1], [2, Appendix 1: “Why exactly FOL”], [6], =-=[21]-=-, [48], [67]. The same reasons motivate the effort of keeping the foundation of spacetime and relativity theory inside FOL. The interplay between logic and relativity theory goes back to around 1920 a... |

6 |
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Citation Context ...thematics was initiated by several authors including David Hilbert [34], cf. also Hilbert’s 6th problem [33], Patrick Suppes [59], Alfred Tarski [32] and leading contemporary logician Harvey Friedman =-=[22, 23]-=-. There are several reasons for seeking an axiomatic foundation of a physical theory [60]. One is that the theory may be better understood by providing a basis of explicit postulates for the theory. A... |

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Citation Context ... situations, [55]. So logic seems to be best suited for providing rational understanding of situations like the circularity represented by CTC’s or time travel. For more on CTC’s, cf. e.g., [15, §6], =-=[17]-=-, [28]. These are only a few of the many examples that show that turning Relativity Theory into a real FOL theory, axiomatizing it and analyzing its logical structure seem to be a promising, worthwile... |

4 |
with contributions from
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Citation Context ... small fragment of the research proposed in sections 1-3. So the proposal is: elaborate a larger part of the perspective outlined in 1-3 in the style of sections 4-5 and [44]. (2) The Introduction of =-=[2]-=- contains more ideas both on the general perspective (of applying logic to spacetime theory) and also more of the long-distance goals. However, some of the present results were not available when [2] ... |

4 |
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Citation Context ...spacetime theory and relativity theories (foundation of relativity) in a spirit analogous with the rather successful foundation of mathematics was initiated by several authors including David Hilbert =-=[34]-=-, cf. also Hilbert’s 6th problem [33], Patrick Suppes [59], Alfred Tarski [32] and leading contemporary logician Harvey Friedman [22, 23]. There are several reasons for seeking an axiomatic foundation... |

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Citation Context ...lativity) in a spirit analogous with the rather successful foundation of mathematics was initiated by several authors including David Hilbert [34], cf. also Hilbert’s 6th problem [33], Patrick Suppes =-=[59]-=-, Alfred Tarski [32] and leading contemporary logician Harvey Friedman [22, 23]. There are several reasons for seeking an axiomatic foundation of a physical theory [60]. One is that the theory may be ... |

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Citation Context ... cf. [44]. Let us call the collection of the axioms introduced so far AccReld: AccReld := SpecReld ∪ { AxEv + , AxSelf + , AxAcc + } ∪ IND. The so-called Twin Paradox is provable in AccRel, cf. [44], =-=[61]-=-. We now formulate the Twin Paradox with our logical notation. The set of events encountered by m ∈ Ob between e1, e2 ∈ Ev is denoted as Evm(e1, e2) := { e ∈ Evm : m ∈ e ∧ Crdm(e1)t < Crdm(e)t < Crdm(... |

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Citation Context ...her-order logic cannot have one by Gödel’s incompleteness theorem, see for example, Väänänen [65, p.505]. For more motivation for staying inside FOL as opposed to higher-order logic, see for example, =-=[1]-=-, [2, Appendix 1: “Why exactly FOL”], [6], [21], [48], [67]. The same reasons motivate the effort of keeping the foundation of spacetime and relativity theory inside FOL. The interplay between logic a... |

3 |
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Citation Context ...o-called Closed Timelike Curves (CTC). Such are Gödel’s rotating universe [25], Kerr and Kerr – Newman spacetimes [47], Gott’s spacetime [27], Tipler’s rotating cylinder [64], van Stockum’s spacetime =-=[57]-=-, Taub-NUT spacetime [31], to mention only a few. Since timelike curves correspond to possible life-lines of observers, in these spacetimes an observer can go through the same event more than once. Th... |

2 |
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Citation Context ...s incompleteness theorem, see for example, Väänänen [65, p.505]. For more motivation for staying inside FOL as opposed to higher-order logic, see for example, [1], [2, Appendix 1: “Why exactly FOL”], =-=[6]-=-, [21], [48], [67]. The same reasons motivate the effort of keeping the foundation of spacetime and relativity theory inside FOL. The interplay between logic and relativity theory goes back to around ... |

2 |
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Citation Context ...better than by changing Einstein’s field equations. This might be useful when we would like to understand the connection of GR to other theories of gravitation like the Brans – Dicke theory, cf. [8], =-=[9]-=-, [20]. This flexibility can also be useful when we would like to extend GR. We indeed would like to extend GR since we do notFOL FOUNDATION OF RELATIVITY THEORIES 6 have a good theory of Quantum Gra... |

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Citation Context ...very leads to many questions and to the idea that the cosmological constant might be replaced with a dynamical parameter, i.e. with a scalar field, called Quintessence or “dark energy” cf. e.g., [10],=-=[13]-=-. But this leads to a new need for modifying or at least fine-tuning GR. This also shows the merit of making GR more flexible by providing a FOL axiom system for it. So far we have talked mainly about... |

2 |
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Citation Context ...t analogous with the rather successful foundation of mathematics was initiated by several authors including David Hilbert [34], cf. also Hilbert’s 6th problem [33], Patrick Suppes [59], Alfred Tarski =-=[32]-=- and leading contemporary logician Harvey Friedman [22, 23]. There are several reasons for seeking an axiomatic foundation of a physical theory [60]. One is that the theory may be better understood by... |

2 |
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Citation Context ...ies (foundation of relativity) in a spirit analogous with the rather successful foundation of mathematics was initiated by several authors including David Hilbert [34], cf. also Hilbert’s 6th problem =-=[33]-=-, Patrick Suppes [59], Alfred Tarski [32] and leading contemporary logician Harvey Friedman [22, 23]. There are several reasons for seeking an axiomatic foundation of a physical theory [60]. One is th... |