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

Giovannini, Eduardo N.

doi:10.1387/theoria.4544

Cited by 5 publications

(9 citation statements)

References 14 publications

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“…However, as Centrone (cf. [9], p. 169) has already showed, other characterizations of definiteness are rather semantical than syntactical 20 :…”

Section: "Absolute Definiteness" As Syntactic Completenessmentioning

confidence: 98%

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Completeness: From Husserl to Carnap

Utrero

2021

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In his Doppelvortrag (1901), Edmund Husserl introduced two concepts of “definiteness” which have been interpreted as a vindication of his role in the history of completeness. Some commentators defended that the meaning of these notions should be understood as categoricity, while other scholars believed that it is closer to syntactic completeness. A detailed study of the early twentieth-century axiomatics and Husserl’s Doppelvortrag shows, however, that many concepts of completeness were conflated as equivalent. Although “absolute definiteness” was principally an attempt to characterize non-extendible manifolds and axiom systems (different from Hilbert’s axiom of completeness), an absolutely definite theory has a unique model and, thus, it is non-forkable and semantically complete (decidable). Non-forkability and decidability were formally delimited by Fraenkel and Carnap almost three decades later and, in fact, they mentioned Husserl as precursor of the latter. Therefore, this paper contributes to a reassessment of Husserl’s place in the history of logic.

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“…However, as Centrone (cf. [9], p. 169) has already showed, other characterizations of definiteness are rather semantical than syntactical 20 :…”

Section: "Absolute Definiteness" As Syntactic Completenessmentioning

confidence: 98%

“…(cf. Giovannini [20], p. 154, fn. 22; my translation) It is thus clear that, in 1897, Hilbert attempted to clarify what is meant by the univocal definition of a "system of objects".…”

Section: "Absolute Definiteness" As Categoricitymentioning

confidence: 99%

Completeness: From Husserl to Carnap

Utrero

2021

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“…The later line-completeness [Hilbert 1962] is a technical variant 53 . Giovannini's account [Giovannini 2013], which relies on [Hallett & Majer 2004] includes a number of points already made here and three more. First, Hilbert's completeness axiom is not about deductive completeness (despite having such consequences), but about maximality of every model (page 145).…”

Section: Hilbert and Dedekind On Continuitymentioning

confidence: 99%

Axiomatizing Changing Conceptions of the Geometric Continuum II: Archimedes-Descartes-Hilbert-Tarski†

Baldwin

2017

Philosophia Mathematica

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In Part I of this paper we argued that the first-order systems HP5 and EG are modest complete descriptive axiomatization of most of Euclidean geometry. In this paper we discuss two further modest complete descriptive axiomatizations: Tarksi's for Cartesian geometry and new systems for adding π. In contrast we find Hilbert's full second order system immodest for geometrical purposes but appropriate as a foundation for mathematical analysis.

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“…Hilbert, 2004Hilbert, [1902, p. 564). 32 Sobre la inclusión del axioma de completitud en ediciones posteriores de Fundamentos de la geometría (1899) puede verse Giovannini (2013).…”

Section: El Cálculo De Segmentos [Streckenrechnungen] De Hilbertunclassified

Aritmetizando la geometría desde dentro: el cálculo de segmentos de David Hilbert

Giovannini¹

2015

Sci. stud.

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resumen Sobre la base que aportan las notas manuscritas de David Hilbert para cursos sobre geometría, el artículo procura contextualizar y analizar una de las contribuciones más importantes y novedosas de su célebre monografía Fundamentos de la geometría (1899), a saber: el cálculo de segmentos lineales (Streckenrechnungen). Se argumenta que, además de ser un resultado matemático importante, Hilbert depositó en su aritmética de segmentos un destacado significado epistemológico y metodológico. En particular, se afirma que para Hilbert este resultado representaba un claro ejemplo de uno de los rasgos más fructíferos y atractivos de su nuevo método axiomático formal, o sea, la capacidad de descubrir y exhibir conexiones estructurales o internas entre diferentes teorías matemáticas.Palabras-clave • Hilbert. Método axiomático. Unidad de la matemática. Geometría euclídea. Aritmetización.

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

Cited by 5 publications

References 14 publications

Completeness: From Husserl to Carnap

Completeness: From Husserl to Carnap

Axiomatizing Changing Conceptions of the Geometric Continuum II: Archimedes-Descartes-Hilbert-Tarski†

Aritmetizando la geometría desde dentro: el cálculo de segmentos de David Hilbert

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