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This note discusses some problems concerning intended, standard, and non- standard models of mathematical theories. We pay attention to the role of extremal axioms in attempts at a unique characterization of the intended models. We recall also Jan Woleński's views on these issues. (original abstract)
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- Adam Mickiewicz University in Poznań, Poland
Bibliografia
- Awodey, S., and E. H. Reck. Completeness and categoricity. Part I: Nineteenth-century axiomatics to twentieth-century metalogic, History and Philosophy of Logic 23, 2002, pp. 1-30.
- Bedurftig, T., and R. Murawski. Philosophy of mathematics, Berlin Boston: Walter de Gruyter GmbH, 2018.
- Borsuk, K., and W. Szmielew. Podstawy geometrii, Warszawa: Państwowe Wydawnictwo Naukowe, 1975.
- Carnap, R., and F. Bachmann. Uber Extremalaxiome, Erkenntnis 6, 1936, pp. 166-188.
- Carnap, R., and F. Bachmann. On extremal axioms [English translation of Carnap and Bachmann 1936, by H.G. Bohnert], History and Philosophy of Logic 2, 1981, pp. 67-85.
- Corcoran, J. Categoricity, History and Philosophy of Logic 1, 1980, pp. 187-207.
- Corcoran, J. From categoricity to completeness, History and Philosophy of Logic 2, 1981, pp. 113-119.
- Feferman, S. Conceptions of the continuum, Intellectica 51, 2009, pp. 169-189.
- Fraenkel, A. A. Einleitung in die Mengenlehre, Berlin: Verlag von Julius Springer, Berlin, 1928.
- Fraenkel, A. A., Y. Bar-Hillel, and A. Levy. Foundations of set theory, Amsterdam London: North-Holland Publishing Company, 1973.
- Gaifman, H. Nonstandard models in a broader perspective, In A. Enayat and R. Kossak (eds.), Nonstandard models in arithmetic and set theory. AMS Special Session Nonstandard Models of Arithmetic and Set Theory, January 15-16, 2003, Baltimore, Maryland. Contemporary Mathematics 361, Providence, Rhode Island: American Mathematical Society, 2004, pp. 1-22.
- Grzegorczyk, A. On the concept of categoricity, Studia Logica 13, 1962, pp. 39-66.
- Hilbert, D. Grundlagen der Geometrie, Festschrift zur Feier der Enthullung des Gauss-Weber- Denkmals in Gottingen, Leipzig: Teubner, 1899.
- Hodges, W. Model theory, Cambridge: Cambridge University Press, 1993.
- Lindenbaum, A., and A. Tarski. Uber die Beschrankheit der Ausdruckmittel deduktiver Theorien, Ergebnisse eines mathematischen Kolloquiums 1934-1935, 7, 1936, pp. 15-22.
- Mancosu, P. The adventure of reason. Interplay between philosophy and mathematical logic, 1900-1940, Oxford: Oxford University Press, 2010.
- Marker, D. Model theory: an introduction, New York Berlin Heidelberg: Springer-Verlag, 2002.
- Murawski, R. Funkcje rekurencyjne i elementy metamatematyki. Problemy zupełności, rozstrzygalności, twierdzenia Godla, Poznań: Wydawnictwo Naukowe UAM, 2000.
- Pogonowski, J. Extremal axioms. Logical, mathematical and cognitive aspects, Poznań: Wydawnictwo Nauk Społecznych i Humanistycznych UAM, 2019.
- Putnam, H. Models and reality, The Journal of Symbolic Logic 45, 1980, pp. 464-482.
- Schiemer, G. Carnap on extremal axioms, 'completeness of models', and categoricity, The Review of Symbolic Logic 5 (4), 2012, pp. 613-641.
- Tarski, A. On the completeness and categoricity of deductive theories. Appendix in Mancosu, P. The adventure of reason. Interplay between philosophy and mathematical logic, 1900-1940, Oxford: Oxford University Press, 2010, 1940, pp. 485-492.
- Woleński, J. Metamatematyka a epistemologia, Warszawa: Wydawnictwo Naukowe PWN, 1993.
- Woleński, J. Kwadrat logiczny - uogólnienia, interpretacje, In J. Perzanowski and A. Pietruszczak (eds.), Logika & filozofia logiczna, Toruń: Wydawnictwo Uniwersytetu Mikołaja Kopernika, 2000, pp. 45-57.
- Woleński, J. Epistemologia. Poznanie, prawda, wiedza, realizm, Warszawa: Wydawnictwo Naukowe PWN, 2005.
- Zermelo, E. Uber Grenzzahlen und Mengenbereiche: Neue Untersuchungen uber die Grundlagen der Mengenlehre, Fundamenta Mathematicae 16, 1930, pp. 29-47.
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