Axiomatic Set Theory à la Dijkstra and Scholten
The algebraic approach by E.W. Dijkstra and C.S. Scholten to formal logic is a proof calculus, where the notion of proof is a sequence of equivalences proved – mainly – by using substitution of ‘equals for equals’. This paper presents Set , a first-order logic axiomatization for set theory using the...
- Autores:
-
Acosta, Ernesto
Aldana, Bernarda
Bohórquez, Jaime
Rocha, Camilo
- Tipo de recurso:
- Part of book
- Fecha de publicación:
- 2017
- Institución:
- Escuela Colombiana de Ingeniería Julio Garavito
- Repositorio:
- Repositorio Institucional ECI
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.escuelaing.edu.co:001/1836
- Acceso en línea:
- https://repositorio.escuelaing.edu.co/handle/001/1836
- Palabra clave:
- Teoría axiomática de conjuntos
Lógica de Dijkstra-Scholten
Manipulación simbólica
SET
Axiomatic set theory
Dijkstra-Scholten logic
Derivation
Formal system
Zermelo-Fraenkel (ZF)
Symbolic manipulation
Undergraduate-level course
- Rights
- closedAccess
- License
- © Springer Nature Switzerland AG 2018
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Axiomatic Set Theory à la Dijkstra and Scholten |
| title |
Axiomatic Set Theory à la Dijkstra and Scholten |
| spellingShingle |
Axiomatic Set Theory à la Dijkstra and Scholten Teoría axiomática de conjuntos Lógica de Dijkstra-Scholten Manipulación simbólica SET Axiomatic set theory Dijkstra-Scholten logic Derivation Formal system Zermelo-Fraenkel (ZF) Symbolic manipulation Undergraduate-level course |
| title_short |
Axiomatic Set Theory à la Dijkstra and Scholten |
| title_full |
Axiomatic Set Theory à la Dijkstra and Scholten |
| title_fullStr |
Axiomatic Set Theory à la Dijkstra and Scholten |
| title_full_unstemmed |
Axiomatic Set Theory à la Dijkstra and Scholten |
| title_sort |
Axiomatic Set Theory à la Dijkstra and Scholten |
| dc.creator.fl_str_mv |
Acosta, Ernesto Aldana, Bernarda Bohórquez, Jaime Rocha, Camilo |
| dc.contributor.author.none.fl_str_mv |
Acosta, Ernesto Aldana, Bernarda Bohórquez, Jaime Rocha, Camilo |
| dc.contributor.researchgroup.spa.fl_str_mv |
Informática |
| dc.subject.armarc.SPA.fl_str_mv |
Teoría axiomática de conjuntos Lógica de Dijkstra-Scholten Manipulación simbólica |
| topic |
Teoría axiomática de conjuntos Lógica de Dijkstra-Scholten Manipulación simbólica SET Axiomatic set theory Dijkstra-Scholten logic Derivation Formal system Zermelo-Fraenkel (ZF) Symbolic manipulation Undergraduate-level course |
| dc.subject.armarc.ENG.fl_str_mv |
SET |
| dc.subject.proposal.eng.fl_str_mv |
Axiomatic set theory Dijkstra-Scholten logic Derivation Formal system Zermelo-Fraenkel (ZF) Symbolic manipulation Undergraduate-level course |
| description |
The algebraic approach by E.W. Dijkstra and C.S. Scholten to formal logic is a proof calculus, where the notion of proof is a sequence of equivalences proved – mainly – by using substitution of ‘equals for equals’. This paper presents Set , a first-order logic axiomatization for set theory using the approach of Dijkstra and Scholten. What is novel about the approach presented in this paper is that symbolic manipulation of formulas is an effective tool for teaching an axiomatic set theory course to sophomore-year undergraduate students in mathematics. This paper contains many examples on how argumentative proofs can be easily expressed in Set and points out how the rigorous approach of Set can enrich the learning experience of students. The results presented in this paper are part of a larger effort to formally study and mechanize topics in mathematics and computer science with the algebraic approach of Dijkstra and Scholten. |
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2017 |
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2017 |
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2021-11-18T13:22:26Z |
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2021-11-18T13:22:26Z |
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Artículo de revista |
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9783319665610 |
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eng |
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eng |
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CCIS;Volumen 735 |
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791 |
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775 |
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Advances in Computing |
| dc.relation.references.spa.fl_str_mv |
Dijkstra, E.W., Scholten, C.S.: Predicate Calculus and Program Semantics. Texts and Monographs in Computer Science. Springer, New York (1990) Halmos, P.R.: Naive Set Theory. Undergraduate Texts in Mathematics. Springer, New York (1974) Hodel, R.E.: An Introduction to Mathematical Logic. Dover Publications Inc., New York (2013) Hrbacek, K., Jech, T.J.: Introduction to Set Theory. Monographs and Textbooks in Pure and Applied Mathematics, vol. 220, 3rd edn. M. Dekker, New York (1999). Rev. and expanded edition Hsiang, J.: Refutational theorem proving using term-rewriting systems. Artif. Intell. 25(3), 255–300 (1985) Jech, T.J.: Set Theory. Pure and Applied Mathematics, a Series of Monographs and Textbooks, vol. 79. Academic Press, New York (1978) Kunen, K.: Set Theory. Studies in Logic, vol. 34. College Publications, London (2013). Revised edition Meseguer, J.: General logics. In: Logic Colloquium 1987: Proceedings. Studies in Logic and the Foundations of Mathematics, 1st edn., vol. 129, pp. 275–330. Elsevier, Granada, August 1989 Meseguer, J.: Conditional rewriting logic as a unified model of concurrency. Theor. Comput. Sci. 96(1), 73–155 (1992) Rocha, C.: The formal system of Dijkstra and Scholten. In: Martí-Oliet, N., Ölveczky, P.C., Talcott, C. (eds.) Logic, Rewriting, and Concurrency. LNCS, vol. 9200, pp. 580–597. Springer, Cham (2015). Rocha, C., Meseguer, J.: A rewriting decision procedure for Dijkstra-Scholten’s syllogistic logic with complements. Revista Colombiana de Computación 8(2), 101–130 (2007) Rocha, C., Meseguer, J.: Theorem proving modulo based on boolean equational procedures. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2008. LNCS, vol. 4988, pp. 337–351. Springer, Heidelberg (2008). Tourlakis, G.J.: Lectures in Logic and Set Theory. Cambridge Studies in Advanced Mathematics, vol. 82–83. Cambridge University Press, Cambridge (2003) |
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© Springer Nature Switzerland AG 2018 |
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Atribución 4.0 Internacional (CC BY 4.0) |
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© Springer Nature Switzerland AG 2018 https://creativecommons.org/licenses/by/4.0/ Atribución 4.0 Internacional (CC BY 4.0) http://purl.org/coar/access_right/c_14cb |
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17 páginas. |
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Springer Nature |
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Acosta, Ernesto6db75777036e0bd7240d41677171e5e6600Aldana, Bernardadd9576ac3ffc19dd2889b032a359ad63600Bohórquez, Jaime34ca64f10c7c3bbadec92bdb453a4170600Rocha, Camilo649eba80a4c919beefa7d19955bc2950600Informática2021-11-18T13:22:26Z2021-11-18T13:22:26Z20179783319665610https://repositorio.escuelaing.edu.co/handle/001/1836The algebraic approach by E.W. Dijkstra and C.S. Scholten to formal logic is a proof calculus, where the notion of proof is a sequence of equivalences proved – mainly – by using substitution of ‘equals for equals’. This paper presents Set , a first-order logic axiomatization for set theory using the approach of Dijkstra and Scholten. What is novel about the approach presented in this paper is that symbolic manipulation of formulas is an effective tool for teaching an axiomatic set theory course to sophomore-year undergraduate students in mathematics. This paper contains many examples on how argumentative proofs can be easily expressed in Set and points out how the rigorous approach of Set can enrich the learning experience of students. The results presented in this paper are part of a larger effort to formally study and mechanize topics in mathematics and computer science with the algebraic approach of Dijkstra and Scholten.El enfoque algebraico de E. W. Dijkstra y C. S. Scholten a la lógica formal es un cálculo de prueba, donde la noción de prueba es una secuencia de equivalencias demostradas, principalmente, mediante la sustitución de "igual por igual". Este artículo presenta Set, una axiomatización lógica de primer orden para la teoría de conjuntos utilizando el enfoque de Dijkstra y Scholten. Lo novedoso del enfoque presentado en este artículo es que la manipulación simbólica de fórmulas es una herramienta eficaz para enseñar un curso de teoría axiomática de conjuntos a estudiantes de segundo año de pregrado en matemáticas. Este documento contiene muchos ejemplos sobre cómo las pruebas argumentativas se pueden expresar fácilmente en Set y señala cómo el enfoque riguroso de Set puede enriquecer la experiencia de aprendizaje de los estudiantes. Los resultados presentados en este artículo son parte de un esfuerzo mayor para estudiar y mecanizar formalmente temas de matemáticas e informática con el enfoque algebraico de Dijkstra y Scholten.17 páginas.application/pdfengSpringer NatureUSA.CCIS;Volumen 735791775N/AAdvances in ComputingDijkstra, E.W., Scholten, C.S.: Predicate Calculus and Program Semantics. Texts and Monographs in Computer Science. Springer, New York (1990)Halmos, P.R.: Naive Set Theory. Undergraduate Texts in Mathematics. Springer, New York (1974)Hodel, R.E.: An Introduction to Mathematical Logic. Dover Publications Inc., New York (2013)Hrbacek, K., Jech, T.J.: Introduction to Set Theory. Monographs and Textbooks in Pure and Applied Mathematics, vol. 220, 3rd edn. M. Dekker, New York (1999). Rev. and expanded editionHsiang, J.: Refutational theorem proving using term-rewriting systems. Artif. Intell. 25(3), 255–300 (1985)Jech, T.J.: Set Theory. Pure and Applied Mathematics, a Series of Monographs and Textbooks, vol. 79. Academic Press, New York (1978)Kunen, K.: Set Theory. Studies in Logic, vol. 34. College Publications, London (2013). Revised editionMeseguer, J.: General logics. In: Logic Colloquium 1987: Proceedings. Studies in Logic and the Foundations of Mathematics, 1st edn., vol. 129, pp. 275–330. Elsevier, Granada, August 1989Meseguer, J.: Conditional rewriting logic as a unified model of concurrency. Theor. Comput. Sci. 96(1), 73–155 (1992)Rocha, C.: The formal system of Dijkstra and Scholten. In: Martí-Oliet, N., Ölveczky, P.C., Talcott, C. (eds.) Logic, Rewriting, and Concurrency. LNCS, vol. 9200, pp. 580–597. Springer, Cham (2015).Rocha, C., Meseguer, J.: A rewriting decision procedure for Dijkstra-Scholten’s syllogistic logic with complements. Revista Colombiana de Computación 8(2), 101–130 (2007)Rocha, C., Meseguer, J.: Theorem proving modulo based on boolean equational procedures. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2008. LNCS, vol. 4988, pp. 337–351. Springer, Heidelberg (2008).Tourlakis, G.J.: Lectures in Logic and Set Theory. Cambridge Studies in Advanced Mathematics, vol. 82–83. Cambridge University Press, Cambridge (2003)© Springer Nature Switzerland AG 2018https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/closedAccessAtribución 4.0 Internacional (CC BY 4.0)http://purl.org/coar/access_right/c_14cbAxiomatic Set Theory à la Dijkstra and ScholtenArtículo de revistainfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_3248http://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/bookParthttp://purl.org/redcol/resource_type/ARThttp://purl.org/coar/version/c_970fb48d4fbd8a85Teoría axiomática de conjuntosLógica de Dijkstra-ScholtenManipulación simbólicaSETAxiomatic set theoryDijkstra-Scholten logicDerivationFormal systemZermelo-Fraenkel (ZF)Symbolic manipulationUndergraduate-level courseORIGINALAxiomatic Set Theory à la Dijkstra and Scholten.pdfAxiomatic Set Theory à la Dijkstra and Scholten.pdfapplication/pdf114783https://repositorio.escuelaing.edu.co/bitstream/001/1836/1/Axiomatic%20Set%20Theory%20%c3%a0%20la%20Dijkstra%20and%20Scholten.pdfce3159b68d11073dc69f09df67ff8a9dMD51open accessLICENSElicense.txtlicense.txttext/plain; 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