Artículos con la etiqueta ‘axiomática’

(In)dependence Logic and Abstract Independence Relations

Por • 30 ene, 2014 • Category: Opinion

We generalize the results of [15] and [16] to the framework of abstract independence relations for an arbitrary AEC. We give a model-theoretic interpretation of the independence atom and characterize under which conditions we can prove a completeness result with respect to the deductive system that axiomatizes independence in team semantics.

Saccheri’s Rectilinear Quadrilaterals

Por • 19 ago, 2013 • Category: Opinion

We study Saccheri’s three hypotheses on a two right-angled isosceles quadrilateral, with a rectilinear summit side. We claim that in the Hilbert`s foundation of geometry the euclidean parallelism is a theorem and as that it can be used, in the hyperbolic geometry

Análisis metamatemático de los números reales

Por • 20 dic, 2012 • Category: Opinion

Postulemos la existencia ideal del conjunto N de todos los números naturales, y, consiguientemente, postulemos que razonar sobre el conjunto N no nos lleva a la contradicción; es fundamental la cuestión siguiente: ¿Se infiere de esa hipótesis la existencia ideal del conjunto potencial {0,1}N, y, por consiguiente, su consistencia lógica?. Sería muy arriesgado, y dejarse engañar por el signo, contestar afirmativamente. […]

On Logical Analysis of Relativity Theories

Por • 27 mar, 2012 • Category: Filosofía

The aim of this paper is to give an introduction to our axiomatic logical analysis of relativity theories.

Resolving Gödel’s Incompleteness Myth: Polynomial Equations and Dynamical Systems for Algebraic Logic

Por • 26 dic, 2011 • Category: Crítica

A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to Goedel’s incompleteness theorems. The truth value of a logical formula subject to a set of axioms is computed from the solution to the corresponding system of polynomial equations. A reference by a formula to its own provability is shown to be a recurrence relation, which can be either interpreted as such to generate a discrete dynamical system, or interpreted in a static way to create an additional simultaneous equation. In this framework the truth values of logical formulas and other polynomial objectives have complex data structures: sets of elementary values, or dynamical systems that generate sets of infinite sequences of such solution-value sets. Besides the routine result that a formula has a definite elementary value, these data structures encode several exceptions: formulas that are ambiguous, unsatisfiable, unsteady, or contingent. These exceptions represent several semantically different types of undecidability; none causes any fundamental problem for mathematics. It is simple to calculate that Goedel’s formula, which asserts that it cannot be proven, is exceptional in specific ways: interpreted statically, the formula defines an inconsistent system of equations (thus it is called unsatisfiable); interpreted dynamically, it defines a dynamical system that has a periodic orbit and no fixed point (thus it is called unsteady). These exceptions are not catastrophic failures of logic; they are accurate mathematical descriptions of Goedel’s self-referential construction. Goedel’s analysis does not reveal any essential incompleteness in formal reasoning systems, nor any barrier to proving the consistency of such systems by ordinary mathematical means.

Towards an axiomatic system for Kolmogorov complexity

Por • 4 feb, 2011 • Category: Filosofía

In this paper, we try to extend this work, both by looking at alternative axiomatic systems for plain complexity and by considering potential axiomatic systems for other types of complexity.