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

  • SEAN WALSH (a1)
Abstract
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.

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[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 ${\rm{\Delta }}_1^1$ -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.
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