164 research outputs found
Adversus singularitates: The ontology of space-time singularities
I argue that there are no physical singularities in space-time. Singular
space-time models do not belong to the ontology of the world, because of a
simple reason: they are concepts, defective solutions of Einstein's field
equations. I discuss the actual implication of the so-called singularity
theorems. In remarking the confusion and fog that emerge from the reification
of singularities I hope to contribute to a better understanding of the
possibilities and limits of the theory of General Relativity.Comment: 13 pages, accepted for publication in Foundations of Scienc
On the Mutual Definability of the Notions of Entailment, Rejection, and Inconsistency
In this paper, two axiomatic theories Tâ and TⲠare constructed, which are dual to Tarskiâs theory T+ (1930) of deductive systems based on classical propositional calculus. While in Tarskiâs theory T+ the primitive notion is the classical consequence function (entailment) Cn+, in the dual theory Tâ it is replaced by the notion of SĹupeckiâs rejection consequence Cnâ and in the dual theory TⲠit is replaced by the notion of the family Incons of inconsistent sets. The author has proved that the theories T+, Tâ, and TⲠare equivalent
Axiomatization and Models of Scientific Theories
In this paper we discuss two approaches to the axiomatization of scien- tific theories in the context of the so called semantic approach, according to which (roughly) a theory can be seen as a class of models. The two approaches are associated respectively to Suppesâ and to da Costa and Chuaquiâs works. We argue that theories can be developed both in a way more akin to the usual mathematical practice (Suppes), in an informal set theoretical environment, writing the set theoretical predicate in the language of set theory itself or, more rigorously (da Costa and Chuaqui), by employing formal languages that help us in writing the postulates to define a class of structures. Both approaches are called internal, for we work within a mathematical framework, here taken to be first-order ZFC. We contrast these approaches with an external one, here discussed briefly. We argue that each one has its strong and weak points, whose discussion is relevant for the philosophical foundations of science
Incompleteness via paradox and completeness
This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various âparadoxical notionsâ for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) andWang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russellâs paradox, a variant of Mirimanoâs paradox, the Liar, and the Grelling-Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth
For Cybersecurity, Computer Science Must Rely on the Opposite of GĂśdelâs Results
This article shows how fundamental higher-order theories of mathematical structures of computer science (e.g. natural numbers [Dedekind 1888] and Actors [Hewitt et. al. 1973]) are cetegorical meaning that they can be axiomatized up to a unique isomorphism thereby removing any ambiguity in the mathematical structures being axiomatized. Having these mathematical structures precisely defined can make systems more secure because there are fewer ambiguities and holes for cyberattackers to exploit. For example, there are no infinite elements in models for natural numbers to be exploited. On the other hand, the 1st-order theories of GĂśdelâs results necessarily leave the mathematical structures ill-defined, e.g., there are necessarily models with infinite integers.
Cyberattackers have severely damaged national, corporate, and individual security as well causing hundreds of billions of dollars of economic damage. A significant cause of the damage is that current engineering practices are not sufficiently grounded in theoretical principles. In the last two decades, little new theoretical work has been done that practically impacts large engineering projects with the result that computer systems engineering education is insufficient in providing theoretical grounding. If the current cybersecurity situation is not quickly remedied, it will soon become much worse because of the projected development of Scalable Intelligent Systems by 2025 [Hewitt 2019].
GĂśdel strongly advocated that the Turing Machine is the preeminent universal model of computation. A Turing machine formalizes an algorithm in which computation proceeds without external interaction. However, computing is now highly interactive, which this article proves is beyond the capability of a Turing Machine. Instead of the Turing Machine model, this article presents an axiomatization of a universal model of digital computation (including implementation of Scalable Intelligent Systems) up to a unique isomorphism
Fuzzy Sets and Formal Logics
The paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics. Our focus is on the historical side of these developments. Š 2015 Elsevier B.V. All rights reserved.partial support by the Spanish projects EdeTRI (TIN2012-39348- C02-01) and 2014 SGR 118.Peer reviewe
First-Order Logic Foundation of Relativity Theories
Motivation and perspective for an exciting new research direction
interconnecting logic, spacetime theory, relativity--including such
revolutionary areas as black hole physics, relativistic computers, new
cosmology--are presented in this paper. We would like to invite the logician
reader to take part in this grand enterprise of the new century. Besides
general perspective and motivation, we present initial results in this
direction.Comment: 25 pages, 4 figure
For Cybersecurity, Computer Science Must Rely on the Opposite of GĂśdelâs Results
This article shows how fundamental higher-order theories of mathematical structures of computer science (e.g. natural numbers [Dedekind 1888] and Actors [Hewitt et. al. 1973]) are cetegorical meaning that they can be axiomatized up to a unique isomorphism thereby removing any ambiguity in the mathematical structures being axiomatized. Having these mathematical structures precisely defined can make systems more secure because there are fewer ambiguities and holes for cyberattackers to exploit. For example, there are no infinite elements in models for natural numbers to be exploited. On the other hand, the 1st-order theories of GĂśdelâs results necessarily leave the mathematical structures ill-defined, e.g., there are necessarily models with infinite integers.
Cyberattackers have severely damaged national, corporate, and individual security as well causing hundreds of billions of dollars of economic damage. A significant cause of the damage is that current engineering practices are not sufficiently grounded in theoretical principles. In the last two decades, little new theoretical work has been done that practically impacts large engineering projects with the result that computer systems engineering education is insufficient in providing theoretical grounding. If the current cybersecurity situation is not quickly remedied, it will soon become much worse because of the projected development of Scalable Intelligent Systems by 2025 [Hewitt 2019].
GĂśdel strongly advocated that the Turing Machine is the preeminent universal model of computation. A Turing machine formalizes an algorithm in which computation proceeds without external interaction. However, computing is now highly interactive, which this article proves is beyond the capability of a Turing Machine. Instead of the Turing Machine model, this article presents an axiomatization of a universal model of digital computation (including implementation of Scalable Intelligent Systems) up to a unique isomorphism
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