Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic

Awodey, Steve; Reck, Erich H.

doi:10.1080/01445340210146889

Cited by 82 publications

(70 citation statements)

References 38 publications

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“…Sobre Dedekind véase Ferreirós (2007, 2009. Un análisis histórico de las nociones de completitud y categoricidad puede verse en Awodey y Reck (2002). 23 Esta sustitución es necesaria dado que, en su versión original, el axioma de completitud supone la validez del axioma de las paralelas.…”

Section: Categoricidad Y El Axioma De Completitudunclassified

Completitud y continuidad en Fundamentos de la Geometría de Hilbert: acerca del Vollständigkeitsaxiom

Giovannini

2013

THEORIA

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RESUMEN: El artículo documenta y analiza las vicisitudes en torno a la incorporación de Hilbert de su famoso axioma de completitud, en el sistema axiomático para la geometría euclídea. Esta tarea es emprendida sobre la base del material que aportan sus notas manuscritas para clases, correspondientes al período [1894][1895][1896][1897][1898][1899][1900][1901][1902][1903][1904][1905]. Se argumenta que este análisis histórico y conceptual no sólo permite ganar claridad respecto de cómo Hilbert concibió originalmente la naturaleza y función del axioma de completitud en su versión geométrica, sino que además permite disipar equívocos en cuanto a la relación de este axioma con la propiedad metalógica de completitud de un sistema axiomático, tal como fue concebida por Hilbert en esta etapa inicial.Descriptores: Hilbert, geometría euclídea, axioma de completitud, axiomas de continuidad, método axiomático, filosofía de la geometría ABSTRACT: The paper reports and analyzes the vicissitudes around Hilbert's inclusion of his famous axiom of completeness, into his axiomatic system for Euclidean geometry. This task is undertaken on the basis of his unpublished notes for lecture courses, corresponding to the period 1894-1905. It is argued that this historical and conceptual analysis not only sheds new light on how Hilbert conceived originally the nature of his geometrical axiom of completeness, but also it allows to clarify some misunderstandings regarding this axiom and the metalogical property of completeness of an axiomatic system, as it was understood by Hilbert in this initial stage.Keywords: Hilbert, Euclidean geometry, axiom of completeness, continuity axioms, axiomatic method, philosophy of geometry * En primer lugar, quisiera agradecer expresamente a los dos árbitros anónimos de Theoria, cuyas observaciones y sugerencias me permitieron introducir importantes mejoras y aclaraciones en la versión original del artículo. Versiones preliminares de este trabajo fueron presentadas en el 16th Brazilian Logic Conference (Petrópolis) y en el Colloqium on Mathematical Logic (ILLC-Amsterdam). Agradezco a los participantes de estas secciones por las críticas realizadas. Desearía además expresar mi agradecimiento a Alejandro Cassini, Ricardo Toledano y Jorge Roetti, por sus comentarios a versiones previas de este escrito. Finalmente agradezco a la Niedersächsische Staats-und Universitätsbi-bliothek Göttingen por el permiso para citar los manuscritos de Hilbert; especialmente al Dr. Helmut Rohlfing, de la Handschriftenabteilung.

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Section: Categoricidad Y El Axioma De Completitudunclassified

Completitud y continuidad en Fundamentos de la Geometría de Hilbert: acerca del Vollständigkeitsaxiom

Giovannini

2013

THEORIA

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“…In his 1929 doctoral dissertation, Gödel proved the completeness theorem for first-order logic, clarifying a relationship between semantic and syntactic notions of logical consequence that had bedeviled early logicians [5]. The dissertation makes frequent references to Hilbert and Ackermann's 1928 Grundzüge der theoretischen Logik, where the problem of proving completeness for first-order logic was articulated clearly.…”

Section: Gödel and The Metamathematical Traditionmentioning

confidence: 99%

Gödel and the metamathematical tradition

Avigad¹

2010

Kurt Gödel

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“…The first half of the twentieth century was a period of great enthusiasm for the recently developed methods of formal logic and formal axiomatics (Awodey and Reck 2002). Pioneers in this tradition-such as Russell, Hilbert, Zermelo, Tarski, Church, Carnap, Quine, and earlier Frege, Peano, and Dedekind, among others-were busy engaging 'hands-on' with projects of formalization concerning specific concepts/topics/areas, in philosophy, mathematics, and beyond.…”

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confidence: 99%

Carnapian explication, formalisms as cognitive tools, and the paradox of adequate formalization

Novaes

Reck

2015

Synthese

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Explication is the conceptual cornerstone of Carnap's approach to the methodology of scientific analysis. From a philosophical point of view, it gives rise to a number of questions that need to be addressed, but which do not seem to have been fully addressed by Carnap himself. This paper reconsiders Carnapian explication by comparing it to a different approach: the 'formalisms as cognitive tools' conception (Formal languages in logic. Cambridge University Press, Cambridge 2012a). The comparison allows us to discuss a number of aspects of the Carnapian methodology, as well as issues pertaining to formalization in general. We start by introducing Carnap's conception of explication, arguing that there is a tension between his proposed criteria of fruitfulness and similarity; we also argue that his further desideratum of exactness is less crucial than might appear at first. We then bring in the general idea of formalisms as cognitive tools, mainly by discussing the reliability of so-called statistical prediction rules (SPRs), i.e. simple algorithms used to make predictions across a range of areas. SPRs allow for a concrete instantiation of Carnap's fruitfulness desideratum, which is arguably the most important desideratum for him. Finally, we elaborate on what we call the 'paradox of adequate formalization', which for the Carnapian corresponds to the tension between similarity and fruitfulness. We conclude by noting that formalization is an inherently paradoxical enterprise in general, but one worth engaging in given the 'cognitive boost' it affords as a tool for discovery.

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Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic

Cited by 82 publications

References 38 publications

Completitud y continuidad en Fundamentos de la Geometría de Hilbert: acerca del Vollständigkeitsaxiom

Completitud y continuidad en Fundamentos de la Geometría de Hilbert: acerca del Vollständigkeitsaxiom

Gödel and the metamathematical tradition

Carnapian explication, formalisms as cognitive tools, and the paradox of adequate formalization

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