F. A. Doria

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 Logics

We expose the main ideas, concepts and results about Jaśkowskis discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

The Incompleteness of Theories of Games

We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.

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.

On Arnold's Hilbert Symposium Problems

We prove that stability is undecidable for dynamical systems whose right-hand side is explicitly written in the language of elementary analysis.

How to build a hypercomputer

We claim that the theoretical hypercomputation problem has already been solved, and that what remains is an engineering problem. We review our construction of the Halting Function (the function that settles the Halting Problem) and then sketch possible blueprints for an actual hypercomputer.

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]:

The Geometry of Gauge Field Copies

We show that a gauge field uniquely determines its potential if and only if its holonomy group coincides with the gauge group on every open set in spacetime, provided that the field is not degenerate as a 2-form over spacetime. In other words, there is no potential ambiguity whenever such a field is irreducible everywhere in spacetime. We then show that the ambiguous potentials for those gauge fields are partitioned into gauge-equivalence classes (modulo certain homotopy classes) as a consequence of the nontrivial connectivity of spacetime. These homotopy classes depend on the gauge group, on the holonomy group and on this last groups centralizer in the gauge group.