Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-20T04:17:42.007Z Has data issue: false hasContentIssue false
  • English
  • Français

CARNAP ON EXTREMAL AXIOMS, “COMPLETENESS OF THE MODELS,” AND CATEGORICITY

Published online by Cambridge University Press:  11 July 2012

GEORG SCHIEMER
GEORG SCHIEMER*
Affiliation:
Munich Center for Mathematical Philosophy, Ludwig-Maximilians-University Munich
*
*MUNICH CENTER FOR MATHEMATICAL PHILOSOPHY LUDWIG-MAXIMILIANS-UNIVERSITY MUNICH 80539 MUNICH, GERMANY Email:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper provides a historically sensitive discussion of Carnap’s theory of extremal axioms developed first in the late 1920s. The main focus is set on the unpublished documents of the projected second part of his manuscript Untersuchungen zur allgemeinen Axiomatik (RC 081-01-01 to 081-01-33). Carnap’s theory will be assessed with respect to two interpretive issues. The first concerns his mathematical sources, that is, the mathematical axioms on which his extremal axioms were based. The second concerns Carnap’s understanding of the relationship between the “completeness of the models” and other metatheoretic notions investigated by him at the time, most notably that of categoricity. The paper surveys Carnap’s different attempts to explicate the extremal properties of a theory and puts his results in context with related metamathematical research at the time.

Type
Research Articles
Information
The Review of Symbolic Logic , Volume 5 , Issue 4 , December 2012 , pp. 613 - 641
Copyright
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Awodey, S. (2010). Category Theory (second edition). Oxford, UK: Oxford University Press.Google Scholar
Awodey, S., & Carus, A. (2001). Carnap, completeness, and categoricity: The Gabelbarkeitssatz of 1928. Erkenntnis, 54, 145–172.CrossRefGoogle Scholar
Awodey, S., & Reck, E. (2002). Completeness and categoricity, part 1: 19th century axiomatics to 20th century metalogic. History and Philosophy of Logic, 23, 1–30.Google Scholar
Baer, R. (1928). Über ein Vollständigkeitsaxiom in der Mengenlehre. Mathematische Zeitschrift, 27, 536–539.Google Scholar
Baldus, R. (1927). Über das Archimedische Axiom. Mathematische Zeitschrift, 26(1), 757–761.Google Scholar
Baldus, R. (1928). Zur Axiomatik der Geometrie. Über Hilberts Vollständigkeitsaxiom. Mathematische Annalen, 100, 321–333.Google Scholar
Bernays, P. (1955). Betrachtungen über das Vollständigkeitsaxiom und verwandte Axiome. Mathematische Zeitschrift, 63, 219–229.Google Scholar
Bonk, T., & Mosterin, J. (2000). Einleitung. In Untersuchungen zur allgemeinen Axiomatik. Darmstadt, Germany: Wissenschaftliche Buchgesellschaft, pp. 1–54.Google Scholar
Carnap, R. (1929). Abriss der Logistik. Wien, Austria: Springer.Google Scholar
Carnap, R. (1930) Bericht über Untersuchungen zur allgemeinen Axiomatik. Erkenntnis, 1, 303–307.Google Scholar
Carnap, R. (1934). Die Antinomien und die Unvollständigkeit der Mathematik. Monatshefte für Mathematik, 41(1), 263–284.Google Scholar
Carnap, R. (1954). Einführung in die Symbolische Logik. Wien, Austria: Springer.Google Scholar
Carnap, R. (2000). Untersuchungen zur allgemeinen Axiomatik. Darmstadt, Germany: Wissenschaftliche Buchgesellschaft.Google Scholar
Carnap, R., & Bachmann, F. (1936). Über Extremalaxiome. Erkenntnis, 6, 166–188. English translation in: History and Philosophy of Logic, 2, 67–85, 1981.Google Scholar
Corcoran, J. (1980). Categoricity. History and Philosophy of Logic, 1, 187–207.Google Scholar
Davey, B. A., & Priestley, H. A. (2002). Introduction to Lattices and Order (second edition). Cambridge, UK: Cambridge University Press.Google Scholar
Dedekind, R. (1888). Was sind und was sollen die Zahlen. Braunschweig, Germany: Vieweg. Page numbers refer to Fricke & Noether (1932).Google Scholar
Demopoulos, W., & Clark, P. (2005). The logicism of Frege, Dedekind, and Russell. In Shapiro, S., editor, The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford, UK: OUP, pp. 129–165.Google Scholar
Ehrlich, P. (1995). Hahn’s Über die Nichtarchimedischen Grössensysteme and the development of the modern theory of magnitudes and numbers to measure them. In Hintikka, J., editor, From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics. Dordrecht, The Netherlands: Kluwer, pp. 165–213.Google Scholar
Ehrlich, P. (1997). From completeness to Archimedean completeness. Synthese, 110, 57–76.Google Scholar
Finsler, P. (1926). Über die Grundlegung der Mengenlehre. Erster Teil. Die Mengen und ihre Axiome. Mathematische Zeitschrift, 25, 683–713.Google Scholar
Fraenkel, A. (1922). Axiomatische Begründung der transfiniten Kardinalzahlen. Mathematische Zeitschrift, 13, 153–188.Google Scholar
Fraenkel, A. (1923). Einleitung in die Mengenlehre (second edition). Berlin: Springer.Google Scholar
Fraenkel, A. (1927). Zehn Vorlesungen über die Grundlegung der Mengenlehre. Leipzig, Germany: Teubner.Google Scholar
Fraenkel, A. (1928) Einleitung in die Mengenlehre (third edition). Berlin: Springer.Google Scholar
Fraenkel, A., & Bar-Hillel, Y. (1958). Foundations of Set Theory. Amsterdam, The Netherlands: North Holland Publishing Company.Google Scholar
Frege, G. (1897). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S., English translation in: van Heijenoort (1967, pp. 1–82).Google Scholar
Fricke, R., & Noether, E., editors. (1932). Richard Dedekind: Gesammelte mathematische Werke, Vol. 3. Braunschweig, Germany: Vieweg & Sohn.Google Scholar
Goldfarb, W. (2005). On Gödel’s way in: the influence of Rudolf Carnap. Bulletin of Symbolic Logic, 11(2), 185–193.Google Scholar
Hilbert, D. (1899). Die Grundlagen der Geometrie. Leipzig, Germany: Teubner.Google Scholar
Hintikka, J. (1991). Carnap, the universality of language and extremality axioms. Erkenntnis, 35, 325–336.Google Scholar
Kanamori, A. (2004). Zermelo and set theory. The Bulletin of Symbolic Logic, 104, 487–553.Google Scholar
Reck, E. (2004). From Frege and Russell to Carnap: Logic and logicism in the 1920s. In Awodey, S., & Klein, C., editors, Carnap Brought Home: The View from Jena. La Salle, IL: Open Court, pp. 151–180.Google Scholar
Reck, E. (2007). Carnap and modern logic. In Friedman, M., & Creath, R., editors, The Cambridge Companion to Carnap. Cambridge, NY: CUP, pp. 176–199.Google Scholar
Schiemer, G. (2010). Fraenkel’s axiom of restriction: axiom choice, intended models and categoricity. In Löwe, B., & Müller, T. editors, Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. London: College Publications, pp. 307–340.Google Scholar
Schiemer, G. (2012). Carnap’s early semantics. Erkenntnis. doi:10.1007/s10670-012-9365-8.Google Scholar
Shapiro, S. (1991). Foundations without Foundationalism - A Case for Second-Order Logic. Oxford, UK: Oxford University Press.Google Scholar
Sieg, W., & Schlimm, D. (2005) Dedekind’s analysis of number: Systems and axioms. Synthese, 147, 121–170.Google Scholar
Smith, P. (2008). Ancestral arithmetic and Isaacsons’s thesis. Analysis, 68, 1–10.CrossRefGoogle Scholar
Torretti, R. (1978). Philosophy of Geometry from Riemann to Poincaré. Dordrecht, Holland: D. Reidel Publishing Co.Google Scholar
van Heijenoort, V., editor. (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge, MA: HUP.Google Scholar
von Neumann, J. (1925) Eine Axiomatisierung der Mengenlehre. Journal für die reine und angewandte Mathematik, 154, 219–240. English translation in van Heijenoort (1967, pp. 183–198).Google Scholar
Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicae, 16, 29–47.Google Scholar