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Galois theory. Structures. Models. Universal theory of structures. Mark Krasner. José Sebastião e Silva.

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COSTA, N. C. A.; BUENO, O. REMARKS ON ABSTRACT GALOIS THEORY. Manuscrito: Revista Internacional de Filosofia, Campinas, SP, v. 34, n. 1, p. 151–183, 2015. Disponível em: Acesso em: 21 fev. 2024.


This paper is a historical companion to a previous one, in which it was studied the so-called abstract Galois theory as formulated by the Portuguese mathematician José Sebastião e Silva (see da Costa, Rodrigues (2007)). Our purpose is to present some applications of abstract Galois theory to higher-order model theory, to discuss Silva’s notion of expressibility and to outline a classical Galois theory that can be obtained inside the two versions of the abstract theory, those of Mark Krasner and of Silva. Some comments are made on the universal theory of (set-theoretic) structures.


BOURBAKI, N. Theory of Sets. London-Ontario: Hermann, Addison Werley, 1968.

BRIGNOLE, D., DA COSTA, N. C. A. “On supernormal Ehresmann-Dedecker universes”, Math. Z., 122, p. 342-350, 1971.

BUNINA, E. I., ZAKHAROV, V. K. “Characterization of model Mirimanov-von Neumann cumulative sets”. J. Math. Sci., 138, p. 5830-5891, 2006.

CAULTON, A., BUTTERFIELD, J. “On kinds of indiscernibil ity in logic and metaphysics” preprint, Cambridge University, 2008.

DA COSTA, N. C. A. “Modéles et univers de Dedecker”, C. R. Acad. Sc. Paris, 275, p. 483-486, 1972.

. Logiques classiques et non classiques. Paris: Masson, 1997.

DA COSTA, N. C. A.; BUENO, O., FRENCH, S. “A coherence theory of truth”, Manuscrito, 29, p. 263-290, 2005.

DA COSTA, N. C. A., RODRIGUES, A. A. M. “Definability and invariance, Studia Logica, 86, p. 1-30, 2007.

DAVEY, B. A., PRIESTLEY, H. A. Introduction to Lattices and Order. Cambridge: Cambridge University Press, 2002.

DENEKE, K., ERNÉ, M., WISMATH, S. L. (eds.) Galois Connections and Applications. Dordrecht: Kluwer Academic Publishers, 2004.

ERNÉ, M., KOSLOWSKI, J., MELTON, A., STRECKER, G. E. “A primer on Galois connections”, Annals of the New York Academy of Sciences, 704, p. 103-125, 1993.

GALATOS, N., JIPSEN, P., KOWALSKI, T., ONO, H. Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Amsterdam: Elsevier, 2007.

GIERZ, G., HOFMANN, K., KEIMEL, K., LAWSON , J., MISLOVE, M., SCOTT, D. Continuous Lattices and Domains. Cambridge: Cambridge University Press, 2002.

KLEIN, F. “Vergleichende Betrachtungen uber neuere geometrische Forschungen”, Mathematische Annalen, 43, p. 63-100, 1893. KRASNER, M. “Une généralisation de la notion de corps”, Journal de Math. Pures et Appl., 17, p. 367-385, 1938.

KLEIN, F. “Généralisation abstraite de la théorie de Galois”, Algébre et Théorie des Nombres, 24, p. 163-168, 1950.

KLEIN, F. “Endothéorie de Galois abstraite”, Séminaire DubreilPisot, Alg. et Théorie des Nombres, 22, 6, 19pp, 1968-69.

KLEIN, F. “Abstract Galois theory”, preprint, University of Paris VI, Paris, 34pp, 1973.

KLEIN, F. “Polythéorie de Galois abstraite dans le cas infini general”, Ann. Sc. Univ. Clermont, Sér. Math., 13, p. 87-91, 1976.

MAC LANE, S. Categories for the Working Mathematician. (Second edition.) New York: Springer, 1998.

MARSHALL, M. V., CHUAQUI, R. “Sentences of type theory: the only sentences preserved under isomorphisms”, Journal of Symbolic Logic, 56, p. 932-948, 1991.

PICADO, J. “The quantale of Galois connections”, Algebra Universalis, 56, p. 527-540, 2005.

ROGERS Jr., H. “Some problems of definability in recursive function theory”, in J. N. Crossley (ed.), Sets, Models and Recursion. Amsterdam: North-Holland, p. 183-201, 1996.

SHOENFIELD, J. R. Mathematical Logic. Reading-London: Addison-Wesley, 1967.

SEBASTIÃO E SILVA, J. “Para Uma Teoria Geral Dos Homomorfismos (Thesis)”, in J. C. Ferreira, J. S. Guerreiro J. S. Oliveira, Obras de José Sebastião e Silva, Volume 1. Lisbon: Sociedade Portuguesa de Matemática, 1985. “Sugli automorfismi di un sistema mathematico qualunque”, Comm. Pontif. Acad. Sci., 9, p. 327-357, 1945.

SEBASTIÃO E SILVA, J. “On automorphisms of arbitrary mathematical systems”, History and Philosophy of Logic, 6, p. 91-116, 1985.

TARSKI, A. Logic, Semantics, Metamathematics. (Edited by John Corcoran.) Indianapolis: Hackett, 1983.

TORRETTI, R. Philosophy of Geometry From Riemann to Poincaré. Dordrecht: Reidel, 1984.

TORRETTI, R. Philosophy of Physics. Cambridge: Cambridge University Press, 1999.

ZAKHAROV, V., BUNINA, E., MIKHALEV, A., ANDREEV, P. “Local theory of sets as a foundation for category theory and its connection with Zermelo-Fraenkel set theory”, J. Math. Sci., 138, p. 5763-5829, 2006.


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