Skip to main content


  • SEAN WALSH (a1)

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.

Hide All
[1] Barwise, Jon, Admissible Sets and Structures, Springer, Berlin, 1975.
[2] Beany, Michael and Reck, Erich H., editors, Gottlob Frege, Critical Assessments of Leading Philosophers, Routledge, London and New York, 2005, Four volumes.
[3] Boolos, George, The iterative conception of set . The Journal of Philosophy, vol. 68 (1971), pp. 215232, Reprinted in [5].
[4] Boolos, George, Iteration again . Philosophical Topics, vol. 17 (1989), pp. 521, Reprinted in [5].
[5] Boolos, George, Logic, Logic, and Logic, Harvard University Press, Cambridge, MA, 1998, Edited by Jeffrey, Richard.
[6] Burgess, John P.,Fixing Frege, Princeton Monographs in Philosophy, Princeton University Press, Princeton, 2005.
[7] Cook, Roy T., Iteration one more time . Notre Dame Journal of Formal Logic, vol. 44 (2003), no. 2, pp. 6392, Reprinted in [8].
[8] 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.
[9] Demopoulos, William, editor, Frege’s Philosophy of Mathematics, Harvard University Press, Cambridge, 1995.
[10] Devlin, Keith J., Constructibility, Perspectives in Mathematical Logic, Springer, Berlin, 1984.
[11] Dummett, Michael, Frege: Philosophy of Mathematics, Harvard University Press, Cambridge, 1991.
[12] Feferman, Solomon, Systems of predicative analysis, this Journal, vol. 29 (1964), pp. 130.
[13] Feferman, Solomon, Predicativity , The Oxford Handbook of Philosophy of Mathematics and Logic (Shapiro, Stewart, editor), Oxford University Press, Oxford, 2005, pp. 590624.
[14] Ferreira, Fernando and Wehmeier, Kai F., On the consistency of the -CA fragment of Frege’s Grundgesetze . Journal of Philosophical Logic, vol. 31 (2002), no. 4, pp. 301311.
[15] Frege, Gottlob, Grundgesetze der Arithmetik: Begriffsschriftlich abgeleitet, Pohle, Jena, 1893, 1903, Two volumes. Reprinted in [16].
[16] Frege, Gottlob, Grundgesetze der Arithmetik: Begriffsschriftlich abgeleitet, Olms, Hildesheim, 1962.
[17] Frege, Gottlob, Basic Laws of Arithmetic, Oxford University Press, Oxford, 2013, Translated by Ebert, Philip A. and Rossberg, Marcus.
[18] 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. 235242.
[19] Gitman, Victoria, Hamkins, Joel David, and Johnstone, Thomas A., What is the theory ZFC without powerset?, arXiv:1110.2430, 2011.
[20] Hale, Bob and Wright, Crispin, The Reason’s Proper Study, Oxford University Press, Oxford, 2001.
[21] 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. 209220.
[22] 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.
[23] Hodes, Harold, Where do sets come from? this Journal, vol. 56 (1991), no. 1, pp. 150175.
[24] Hodges, Wilfrid, Model Theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.
[25] Jech, Thomas, Set Theory, Springer Monographs in Mathematics, Springer, Berlin, 2003, The Third Millennium Edition.
[26] Björn Jensen, R., The fine structure of the constructible hierarchy . Annals of Mathematical Logic, vol. 4 (1972), pp. 229308.
[27] Kechris, Alexander S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer, New York, 1995.
[28] Kripke, Saul, Transfinite recursion on admissible ordinals I, II, this Journal, vol. 29 (1964), no. 3, pp. 161162.
[29] Kunen, Kenneth, Set Theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1980.
[30] Kunen, Kenneth, Set Theory, College Publications, London, 2011.
[31] 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. 161168, Reprinted in [9].
[32] Alan Platek, Richard, Foundations of Recursion Theory, Unpublished Dissertation, Stanford University, Stanford, CA, 1966.
[33] Sacks, Gerald E., Higher Recursion Theory, Perspectives in Mathematical Logic, Springer, Berlin, 1990.
[34] Schindler, Ralf and Zeman, Martin, Fine structure , Handbook of Set Theory (Foreman, Matthew and Kanamori, Akihiro, editors), vol. 1, Springer, Berlin, 2010, pp. 605656.
[35] Shoenfield, Joseph R., The problem of predicativity , Essays on the foundations of mathematics, Magnes Press, Jerusalem, 1961, pp. 132139.
[36] Shoenfield, Joseph R., Chapter 9: Set theory , Mathematical Logic, Addison-Wesley, Reading, 1967, pp. 238315.
[37] 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.
[38] 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. 355390.
[39] Simpson, Stephen G., Subsystems of Second Order Arithmetic, second edition, Cambridge University Press, Cambridge, 2009.
[40] Walsh, Sean, Comparing Hume’s Principle, Basic Law V and Peano arithmetic . Annals of Pure and Applied Logic, vol. 163 (2012), pp. 16791709.
[41] Walsh, Sean, The strength of predicative abstraction, arXiv:1407.3860, 2014.
[42] Walsh, Sean, Predicativity, the Russell-Myhill paradox, and Church’s intensional logic . Journal of Philosophical Logic, forthcoming.
[43] Weyl, Hermann, Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis., Veit, Leipzig, 1918.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 49 *
Loading metrics...

Abstract views

Total abstract views: 254 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th August 2018. This data will be updated every 24 hours.