N. C. A. da Costa

Undecidability and incompleteness in classical mechanics

We describe Richardsons functor from the Diophantine equations and Diophantine problems into elementary real-valued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamiltonian mechanics and dynamical systems theory. Our examples deal with the algorithmic impossibility of deciding whether a given Hamiltonian can be integrated by quadratures and related questions; they lead to a version of Gödels incompleteness theorem within Hamiltonian mechanics. A similar application to the unsolvability of the decision problem for chaotic dynamical systems is also obtained.

Classical physics and Penrose's thesis

We expose and discussPenroses thesis: “Nature produces harnessable noncomputable processes, but none at the classical level.” We then suggest a partial counterexample to it, based on aGedanken experiment about an undecidable family of integrable Hamiltonian systems that could lead to a sort of idealized solution to the Halting problem for Turing machines.

On JaŚkowski's Discussive Logic(*)

Publisher Summary This chapter focuses on Jaśkowskis discussive propositional calculus to a higher-order discussive logic and presents a new axiomatization of discussive propositional calculus that constitutes a solution to Problems I and II of Costa 1975. The discussive propositional calculus has a logical and mathematical importance by itself and has originated several interesting problems. In an axiomatization for the discussive propositional calculus the primitive connectives are: → (discussive implication), Λ (left discussive conjunction), V (disjunction), and ┐ (negation). The formulas of the discussive propositional calculus are constructed with propositional variables, the connectives and parentheses. In order to facilitate the writing of formulas, small Latin letters is employed as syntactical variables. The usual rules of the positive propositional logic are valid in the discussive propositional calculus. The chapter discusses the deduction theorem, Godels incompleteness theorem, various other theorem, axiom schemata, derivatives rules, and various lemmas.

Dynamical system where proving chaos is equivalent to proving Fermat's conjecture

We prove that we can explicitly construct the expression for a low-dimensional Hamiltonian system where proving the existence of a Smale horseshoe is equivalent to proving that Fermats Conjecture is true. We then show that some sets of similar intractable problems are dense (in the usual topology) in the space of all dynamical systems over a finite-dimensional real manifold.

A Suppes predicate for general relativity and set-theoretically generic spacetimes

We summarize ideas from Zermelo-Fraenkel set theory up to an axiomatic treatment for general relativity based on a Suppes predicate. We then examine the meaning of set-theoretic genericity for manifolds that underlie the Einstein equations. A physical interpretation is finally offered for those set-theoretically generic manifolds in gravitational theory.

Structures, Suppes Predicates, and Boolean-Valued Models in Physics

An old and important question concerning physical theories has to do with their axiomatization [47]. The sixth problem in Hilbert’s celebrated list of mathematical problems deals with its desirable (or ideal) contours [31]:

Undecidable hopf bifurcation with undecidable fixed point

We exhibit a polynomial dynamical system where one cannot decide whether a Hopf bifurcation occurs. Therefore one cannot decide whether there will be parameter values such that a stable fixed point becomes an unstable one. Related incompleteness results for previously described axiomatized versions of dynamical systems theory are also discussed.

Some thoughts on hypercomputation

We first show that the Halting Function (the noncomputable function that solves the Halting Problem) has explicit expressions in the language of calculus. Out of that fact we elaborate on the possible meaning of hypercomputation theory within the setting of formal mathematical theories.

Gödel incompleteness, explicit expressions for complete arithmetic degrees and applications

We summarize in an intuitive vein a few concepts from recursion theory and from the theory of formal systems and then state and comment our recent results on the incompleteness of elementary real analysis and its consequences. Their relation to forcing is also dealt with.

Two Questions on the Geometry of Gauge Fields

We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for higher-degree similar relations. We then prove that there is no algorithmic procedure to determine, for a reasonable restricted algebra of functions on spacetime, whether a given connection form satisfies the preceding conditions. A parallel result gives a version of Gödels first incompleteness theorem within an (axiomatized) theory of gauge fields.