Undecidability and incompleteness in classical mechanics

Costa, Newton C. A. da; Dória, Francisco Antônio

doi:10.1007/bf00671484

Cited by 102 publications

(67 citation statements)

References 37 publications

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“…That result led to several undecidability and incompleteness results in dynamical systems theory; all stem from Gödel's original incompleteness theorem for arithmetic through another of the Hilbert problems, the 10th Problem. Actually such examples of incompleteness and undecidability all stem from a very general Rice-like theorem proved by da Costa and Doria (1991a). However, in order to have incompleteness in physics, we must have an axiomatic framework.…”

Section: Axiomatics For Physics: Preliminary Stepsmentioning

confidence: 99%

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Axiomatics, the Social Sciences, and the Gödel Phenomenon: A Toolkit

Dória¹

2017

The Limits of Mathematical Modeling in the Social Sciences

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Section: Axiomatics For Physics: Preliminary Stepsmentioning

confidence: 99%

“…We have used here a variant of the construction of θ and β which first appeared in da Costa and Doria (1991a). Then, we have the following corollary.…”

Section: Ff)mentioning

confidence: 99%

Axiomatics, the Social Sciences, and the Gödel Phenomenon: A Toolkit

Dória¹

2017

The Limits of Mathematical Modeling in the Social Sciences

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“…Fifth, the 'proof'of Cole's result is almost scandalously short. Finally, how Cole (and Kraïchik) arrived at their results seem to be a non-issue; it may or may not have been hard 14 . On the other hand, 1 1 A number is said to be perfect if it is equal to the sum of all its proper divisors (eg., 6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + +7 + 14).…”

Section: Problem 6 Sat -The Satis…ability Problemmentioning

confidence: 99%

“…The point to note is the square root term in (14). No underlying model of computation which is not capable of precise real number computation will be able to implement the ellipsoid algorithm consistently.…”

Section: Remark 17mentioning

confidence: 99%

A Computable Economist’s Perspective on Computational Complexity

Velupillai¹

2009

Handbook of Research on Complexity

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have been educating me, from di¤erent points of view, on many fascinating visions of the thorny concept of complexity. More than a decade and a half of constant, stimulating, discussions with John McCall on Kolmogorovalgorithmic -complexity and theories of probability have been a source of great pleasure and consistent intellectual adventures. Not being particularly competent in the actual programming of theoretically structured computing models, I have had to rely on Stefano Zambelli for instruction and advice on the felicitous link between theory and application that is the hall mark of computational complexity theory. More recently, I have had the pleasure and privilege of being able to pick Chico Doria's fertile brain for exotic ideas about the P =?N P question. Years ago, now bordering on almost two decades, the decision by Shu-Heng Chen to write his UCLA dissertation on the application of algorithmic and stochastic complexity theories, with me as his main thesis advisor, forced me to try to be at least one-step ahead of his incredible learning abilities. Barkley Rosser's stimulating visit to Trento last year provided much inspiration on many topics related to a broader perspective on complexity. I wish I could blame them, and not have to plead guilty, for the infelicities and errors that remain! 1 Abstract A computable economist's view of the world of computational complexity theory is described. This means the model of computation underpinning theories of computational complexity plays a central role. The emergence of computational complexity theories from diverse traditions is emphasised. The uni…ca-tions that emerged in the modern era was codi…ed by means of the notions of e¢ ciency of computations, non-deterministic computations, completeness, reducibility and veri…ability -all three of the latter concepts had their origins on what may be called 'Post's Program of Research for Higher Recursion Theory'. Approximations, computations and constructions are also emphasised. The recent real model of computation as a basis for studying computational complexity in the domain of the reals is also presented and discussed, albeit critically. A brief sceptical section on algorithmic complexity theory is included in an appendix. 21 Prologue "There are many levels of complexity in problems, and corresponding boundaries between them. Turing computability is an outer boundary, ... any theory that requires more power than that surely is irrelevant to any useful de…nition of human rationality. A slightly stricter boundary is posed by computational complexity, especially in its common "worst case" form. We cannot expect people (and/or computers) to …nd exact solutions for large problems in computationally complex domains. This still leaves us far beyond what people and computers actually CAN do. The next boundary... is computational complexity for the "average case" .... .. That begins to bring us closer to the realities of real-world and realtime computation. Finally, we get to the empirical boundary, measured by labora...

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“…For these systems there exists in general no constructive spectral theory [12]. It may even be shown that the solution of the spectral problem then becomes undecidable in the sense of Gödel's theory [13].…”

Section: Poincare's Divergencesmentioning

confidence: 99%

From Poincare's Divergences to Quantum Mechanics with Broken Time Symmetry

Prigogine

1997

Zeitschrift Für Naturforschung A

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We discuss the spectral property of unstable dynamical systems in both classical and quantum mechanics. An important class of unstable dynamical systems corresponds to the Large Poincare Systems (LPS). Conventional perturbation technique leads then to divergences. We introduce methods for the elimination of Poincare divergences to obtain a solution of the spectral problem analytic in the coupling constant. To do so, we have to enlarge the class of permissible transformations, to include non-unitary transformations as well as to extend the Hilbert space. A simple example refers to the Friedrichs model, which was studied independently by George Sudarshan and his co-workers. However, our main interest is the irreducible representations in the Liouville space. In these representations the central quantity is the density matrix, and the eigenfunctions of the Liouville operator cannot be expressed in terms of the wave functions. We suggest that this situation corresponds to quantum chaos. Indeed, classical chaos does not mean that Newton's equation becomes "wrong" but that trajectories loose their operational meaning. Similarly, whenever we have an irreducible representation in the Liouville space this means that the wave function description looses its operational meaning. Additional statistical features appear. A simple example corresponds to persistent interactions in the scattering problem which cannot be treated in the frame of usual S-matrix theory.

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Undecidability and incompleteness in classical mechanics

Cited by 102 publications

References 37 publications

Axiomatics, the Social Sciences, and the Gödel Phenomenon: A Toolkit

Axiomatics, the Social Sciences, and the Gödel Phenomenon: A Toolkit

A Computable Economist’s Perspective on Computational Complexity

From Poincare's Divergences to Quantum Mechanics with Broken Time Symmetry

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