Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T23:39:39.757Z Has data issue: false hasContentIssue false
  • English
  • Français

FRAGMENTS OF FREGE’S GRUNDGESETZE AND GÖDEL’S CONSTRUCTIBLE UNIVERSE

Published online by Cambridge University Press:  29 June 2016

SEAN WALSH
SEAN WALSH*
Affiliation:
DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE 5100 SOCIAL SCIENCE PLAZA UNIVERSITY OF CALIFORNIA, IRVINE IRVINE, CA 92697-5100, USAE-mail:[email protected] or [email protected]: http://www.swalsh108.org
Get access Rights & Permissions [Opens in a new window]

Abstract

Frege’s Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem (Theorem 2.9) shows that there is a model of a fragment of the Grundgesetze which defines a model of all the axioms of Zermelo–Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to Gödel’s constructible universe of sets and to Kripke and Platek’s idea of the projectum, as well as to a weak version of uniformization (which does not involve knowledge of Jensen’s fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension (Theorem 3.5). As an application, we resolve an analogue of the joint consistency problem in the predicative setting.

Keywords

FregeGrundgesetzeconstructibilityabstraction principles
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 2 , June 2016 , pp. 605 - 628
Copyright
Copyright © The Association for Symbolic Logic 2016 

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

REFERENCES

Barwise, Jon, Admissible Sets and Structures, Springer, Berlin, 1975.Google Scholar
Beany, Michael and Reck, Erich H., editors, Gottlob Frege, Critical Assessments of Leading Philosophers, Routledge, London and New York, 2005, Four volumes.Google Scholar
Boolos, George, The iterative conception of set . The Journal of Philosophy, vol. 68 (1971), pp. 215–232, Reprinted in [5].Google Scholar
Boolos, George, Iteration again . Philosophical Topics, vol. 17 (1989), pp. 5–21, Reprinted in [5].Google Scholar
Boolos, George, Logic, Logic, and Logic, Harvard University Press, Cambridge, MA, 1998, Edited by Jeffrey, Richard.Google Scholar
Burgess, John P.,Fixing Frege, Princeton Monographs in Philosophy, Princeton University Press, Princeton, 2005.Google Scholar
Cook, Roy T., Iteration one more time . Notre Dame Journal of Formal Logic, vol. 44 (2003), no. 2, pp. 63–92, Reprinted in [8].Google Scholar
Cook, Roy T., editor, The Arché Papers on the Mathematics of Abstraction, The Western Ontario Series in Philosophy of Science, vol. 71, Springer, Berlin, 2007.Google Scholar
Demopoulos, William, editor, Frege’s Philosophy of Mathematics, Harvard University Press, Cambridge, 1995.Google Scholar
Devlin, Keith J., Constructibility, Perspectives in Mathematical Logic, Springer, Berlin, 1984.CrossRefGoogle Scholar
Dummett, Michael, Frege: Philosophy of Mathematics, Harvard University Press, Cambridge, 1991.Google Scholar
Feferman, Solomon, Systems of predicative analysis, this Journal, vol. 29 (1964), pp. 1–30.Google Scholar
Feferman, Solomon, Predicativity , The Oxford Handbook of Philosophy of Mathematics and Logic (Shapiro, Stewart, editor), Oxford University Press, Oxford, 2005, pp. 590–624.Google Scholar
Ferreira, Fernando and Wehmeier, Kai F., On the consistency of the ${\rm{\Delta }}_1^1$ -CA fragment of Frege’s Grundgesetze . Journal of Philosophical Logic, vol. 31 (2002), no. 4, pp. 301–311.Google Scholar
Frege, Gottlob, Grundgesetze der Arithmetik: Begriffsschriftlich abgeleitet, Pohle, Jena, 1893, 1903, Two volumes. Reprinted in [16].Google Scholar
Frege, Gottlob, Grundgesetze der Arithmetik: Begriffsschriftlich abgeleitet, Olms, Hildesheim, 1962.Google Scholar
Frege, Gottlob, Basic Laws of Arithmetic, Oxford University Press, Oxford, 2013, Translated by Ebert, Philip A. and Rossberg, Marcus.Google Scholar
Friedman, Harvey M., Some systems of second-order arithmetic and their use , Proceedings of the International Congress of Mathematicians, Vancouver 1974, vol. 1, 1975, pp. 235–242.Google Scholar
Gitman, Victoria, Hamkins, Joel David, and Johnstone, Thomas A., What is the theory ZFC without powerset?, arXiv:1110.2430, 2011.Google Scholar
Hale, Bob and Wright, Crispin, The Reason’s Proper Study, Oxford University Press, Oxford, 2001.Google Scholar
Heck, Richard G. Jr., The consistency of predicative fragments of Frege’s Grundgesetze der Arithmetik . History and Philosophy of Logic, vol. 17 (1996), no. 4, pp. 209–220.Google Scholar
Heinzmann, Gerhard, Poincaré, Russell, Zermelo et Peano. Textes de la discussion (1906-1912) sur les fondements des mathématiques: Des antinomie à la prédicativié, Blanchard, Paris, 1986.Google Scholar
Hodes, Harold, Where do sets come from? this Journal, vol. 56 (1991), no. 1, pp. 150–175.Google Scholar
Hodges, Wilfrid, Model Theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.Google Scholar
Jech, Thomas, Set Theory, Springer Monographs in Mathematics, Springer, Berlin, 2003, The Third Millennium Edition.Google Scholar
Björn Jensen, R., The fine structure of the constructible hierarchy . Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.Google Scholar
Kechris, Alexander S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer, New York, 1995.Google Scholar
Kripke, Saul, Transfinite recursion on admissible ordinals I, II, this Journal, vol. 29 (1964), no. 3, pp. 161–162.Google Scholar
Kunen, Kenneth, Set Theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1980.Google Scholar
Kunen, Kenneth, Set Theory, College Publications, London, 2011.Google Scholar
Parsons, Terence, On the consistency of the first-order portion of Frege’s logical system . Notre Dame Journal of Formal Logic, vol. 28 (1987), no. 1, pp. 161–168, Reprinted in [9].Google Scholar
Alan Platek, Richard, Foundations of Recursion Theory, Unpublished Dissertation, Stanford University, Stanford, CA, 1966.Google Scholar
Sacks, Gerald E., Higher Recursion Theory, Perspectives in Mathematical Logic, Springer, Berlin, 1990.Google Scholar
Schindler, Ralf and Zeman, Martin, Fine structure , Handbook of Set Theory (Foreman, Matthew and Kanamori, Akihiro, editors), vol. 1, Springer, Berlin, 2010, pp. 605–656.Google Scholar
Shoenfield, Joseph R., The problem of predicativity , Essays on the foundations of mathematics, Magnes Press, Jerusalem, 1961, pp. 132–139.Google Scholar
Shoenfield, Joseph R., Chapter 9: Set theory , Mathematical Logic, Addison-Wesley, Reading, 1967, pp. 238–315.Google Scholar
Shoenfield, Joseph R., Axioms of set theory , Handbook of Mathematical Logic (Barwise, Jon, editor), Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland, Amsterdam, 1977.CrossRefGoogle Scholar
Simpson, Stephen G., Short course on admissible recursion theory , Generalized recursion theory II, Studies in Logic and the Foundations of Mathematics, vol. 94, North-Holland, Amsterdam, 1978, pp. 355–390.Google Scholar
Simpson, Stephen G., Subsystems of Second Order Arithmetic, second edition, Cambridge University Press, Cambridge, 2009.Google Scholar
Walsh, Sean, Comparing Hume’s Principle, Basic Law V and Peano arithmetic . Annals of Pure and Applied Logic, vol. 163 (2012), pp. 1679–1709.Google Scholar
Walsh, Sean, The strength of predicative abstraction, arXiv:1407.3860, 2014.Google Scholar
Walsh, Sean, Predicativity, the Russell-Myhill paradox, and Church’s intensional logic . Journal of Philosophical Logic, forthcoming.Google Scholar
Weyl, Hermann, Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis., Veit, Leipzig, 1918.Google Scholar