Suppes Predicates and the Construction of Unsolvable Problems in the Axiomatized Sciences

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

doi:10.1007/978-94-011-0776-1_7

Cited by 15 publications

(15 citation statements)

References 23 publications

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“…For those questions, there is no general algorithm available since T contains the sine function, the absolute value function and π; also the corresponding theory is incomplete (see da Costa and Doria, 1994a). In the polynomial case again, there is no algorithm to decide whether a fixed point at the origin is stable or not (see da Costa and Doria, 1993a,b).…”

Section: Let a Vector Field Be Given By Polynomials Of A Fixed Degreementioning

confidence: 99%

“…Lewis and Inagaki (1991b) pointed out that our results entail the incompleteness of the theory of Hamiltonian models in economics. They also entail the incompleteness of the theory of Arrow-Debreu equilibria and (what is at first sight surprising) the incompleteness of the theory of finite games with Nash equilibria (see da Costa and Doria, 1994a;Tsuji et al, 1998). Those two last questions are discussed below in Section 17.…”

Section: Let a Vector Field Be Given By Polynomials Of A Fixed Degreementioning

confidence: 99%

“…Also we can explicitly construct "natural"-looking and quite simple problems with our techniques that lie beyond the arithmetical hierarchy (see da Costa and Doria, 1994a). For example, we can explicitly define a procedure to obtain an expression θ (ω) (n) for a characteristic function in 0 (ω) .…”

Section: Simple Problems Worse Than Any Number-theoretic Problemmentioning

confidence: 99%

“…However, there are no explicit expressions for those objects. With the help of our techniques, we exhibited an expression for a "faceless" Hamiltonian (see da Costa and Doria, 1994a): the only thing we can prove about it is that it definitely is a Hamiltonian, and nothing more.…”

Section: Generic Faceless Objectsmentioning

confidence: 99%

“…Yet the stronger axioms may also be debatable on philosophical grounds, so that the proof of a desired fact from the enriched system eventually turns out to be technically correct but philosophically (and perhaps empirically) doubtful. For details see da Costa and Doria (1994a) and Tsuji et al (1998).…”

Section: Markets In Equilibrium May Have Noncomputable Pricesmentioning

confidence: 99%

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

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2017

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Section: Let a Vector Field Be Given By Polynomials Of A Fixed Degreementioning

confidence: 99%

Section: Let a Vector Field Be Given By Polynomials Of A Fixed Degreementioning

confidence: 99%

Section: Simple Problems Worse Than Any Number-theoretic Problemmentioning

confidence: 99%

Section: Generic Faceless Objectsmentioning

confidence: 99%

Section: Markets In Equilibrium May Have Noncomputable Pricesmentioning

confidence: 99%

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Gödel incompleteness, explicit expressions for complete arithmetic degrees and applications

Costa

Dória²

1995

Complexity

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We summarize in a n intuitiue vein a few concepts froin recursion theory and fi-om the theory ofjormal systems and then state and comment our recent 1-esults on the incoinpleteness of elementary real analysis and its consequences. Their relation to forcing is also dealt with. 0 1995 John Wiiviley & Sons, Inc. a MOTIVATIONhere are simple noncomputable expressions for functions within mathematics; those expressions creep up T even within languages close to arithmetic, and lead to very simple questions such as, "does the integer-valued function O(n) equal 0 or 12" which turn out to be undecidable in the general case. Analogous nai've looking but intractable expressions for functions can also be found within more elaborate languages, as classical elementary analysis. With their help we can generate infinitely many undecidable sentences with a trivial appearance from arithmetic on and all the way up to the whole of mathematics.Some of those intractable expressions represent the halting function e (m, n), that tells us whether the Turing machine M,,,(n) stops over its input n. Once we have an expression for the halting function, we can obtain explicit expressions for all complete arithmetic degrees and even beyond. Therefore, the associated undecidable predicates represent problems in all the corresponding degrees of unsolvability, both inside and outside the arithmetic hierarchy.We used those undecidable predicates and noncomputable functions to settle several open decision problems, mainly in dynamical systems theory. Two such problems deal with frequently handled questions: 0Is there a decision procedure for chaos? Chaos theory has been a fast growing research area since the early 1970s, a decade after the discovery by E. Lorenz of an apparent, but still unproved chaotic behavior in a deterministic nonlinear dynamical system (for references see [7]). Chaos scientists usually proceed in one of two ways: whenever they wish to know if a given physical process is chaotic the usual starting point is to write down the equations that describe the process; out of them one tries to check whether the process satisfies some of the established mathematical criteria for chaos and randomness.However, those equations are in most cases intractable nonlinear differential equations as they cannot in general be given explicit analytical solutions. Therefore, chaos theorists turn to computer simulations and for most non-

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The Significance of Relativistic Computation for the Philosophy of Mathematics

Wójtowicz

2021

Hajnal Andréka and István Németi on Unity of Science

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Suppes Predicates and the Construction of Unsolvable Problems in the Axiomatized Sciences

Cited by 15 publications

References 23 publications

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

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

Gödel incompleteness, explicit expressions for complete arithmetic degrees and applications

The Significance of Relativistic Computation for the Philosophy of Mathematics

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