A Note on Intended and Standard Models

Pogonowski, Jerzy

doi:10.2478/sh-2020-0032

Abstract

This note discusses some problems concerning intended, standard, and nonstandard 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.

References

1. 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. 1-30.10.1080/01445340210146889
Search in Google Scholar Back to article
2. Bedürftig, T., and R. Murawski. Philosophy of mathematics, Berlin Boston: Walter de Gruyter GmbH, 2018.10.1515/9783110468335
Search in Google Scholar Back to article
3. Borsuk, K., and W. Szmielew. Podstawy geometrii, Warszawa: Państwowe Wydawnictwo Naukowe, 1975.
Search in Google Scholar Back to article
4. Carnap, R., and F. Bachmann. Über Extremalaxiome, Erkenntnis 6, 1936, pp. 166-188.10.1007/BF02538231
Search in Google Scholar Back to article
5. 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. 67-85.10.1080/01445348108837022
Search in Google Scholar Back to article
6. Corcoran, J. Categoricity, History and Philosophy of Logic 1, 1980, pp. 187-207.10.1080/01445348008837010
Search in Google Scholar Back to article
7. Corcoran, J. From categoricity to completeness, History and Philosophy of Logic 2, 1981, pp. 113-119.10.1080/01445348108837024
Search in Google Scholar Back to article
8. Feferman, S. Conceptions of the continuum, Intellectica 51, 2009, pp. 169-189.10.3406/intel.2009.1737
Search in Google Scholar Back to article
9. Fraenkel, A. A. Einleitung in die Mengenlehre, Berlin: Verlag von Julius Springer, Berlin, 1928.10.1007/978-3-662-42029-4
Search in Google Scholar Back to article
10. Fraenkel, A. A., Y. Bar-Hillel, and A. Levy. Foundations of set theory, Amsterdam London: North-Holland Publishing Company, 1973.
Search in Google Scholar Back to article
11. 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. 1-22.10.1090/conm/361/06585
Search in Google Scholar Back to article
12. Grzegorczyk, A. On the concept of categoricity, Studia Logica 13, 1962, pp. 39-66.10.1007/BF02317255
Search in Google Scholar Back to article
13. Hilbert, D. Grundlagen der Geometrie, Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen, Leipzig: Teubner, 1899.
Search in Google Scholar Back to article
14. Hodges, W. Model theory, Cambridge: Cambridge University Press, 1993.
Search in Google Scholar Back to article
15. Lindenbaum, A., and A. Tarski. Über die Beschränkheit der Ausdruckmittel deduktiver Theorien, Ergebnisse eines mathematischen Kolloquiums 1934–1935, 7, 1936, pp. 15-22.
Search in Google Scholar Back to article
16. Mancosu, P. The adventure of reason. Interplay between philosophy and mathematical logic, 1900–1940, Oxford: Oxford University Press, 2010.
Search in Google Scholar Back to article
17. Marker, D. Model theory: an introduction, New York Berlin Heidelberg: Springer-Verlag, 2002.
Search in Google Scholar Back to article
18. Murawski, R. Funkcje rekurencyjne i elementy metamatematyki. Problemy zupełności, rozstrzygalności, twierdzenia Gödla, Poznań: Wydawnictwo Naukowe UAM, 2000.
Search in Google Scholar Back to article
19. Pogonowski, J. Extremal axioms. Logical, mathematical and cognitive aspects, Poznań: Wydawnictwo Nauk Społecznych i Humanistycznych UAM, 2019.
Search in Google Scholar Back to article
20. Putnam, H. Models and reality, The Journal of Symbolic Logic 45, 1980, pp. 464-482.10.2307/2273415
Search in Google Scholar Back to article
21. Schiemer, G. Carnap on extremal axioms, ‘completeness of models’, and categoricity, The Review of Symbolic Logic 5 (4), 2012, pp. 613-641.10.1017/S1755020312000172
Search in Google Scholar Back to article
22. 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.
Search in Google Scholar Back to article
23. Woleński, J. Metamatematyka a epistemologia, Warszawa: Wydawnictwo Naukowe PWN, 1993.
Search in Google Scholar Back to article
24. 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.
Search in Google Scholar Back to article
25. Woleński, J. Epistemologia. Poznanie, prawda, wiedza, realizm, Warszawa: Wydawnictwo Naukowe PWN, 2005.
Search in Google Scholar Back to article
26. Zermelo, E. Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae 16, 1930, pp. 29-47.10.4064/fm-16-1-29-47
Search in Google Scholar Back to article

A Note on Intended and Standard Models

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