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THE STRENGTH OF ABSTRACTION WITH PREDICATIVE COMPREHENSION

Published online by Cambridge University Press:  29 March 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: swalsh108@gmail.com or walsh108@uci.eduURL: http://www.walsh108.org
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Abstract

Frege’s theorem says that second-order Peano arithmetic is interpretable in Hume’s Principle and full impredicative comprehension. Hume’s Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege’s Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic (cf. Theorem 3.2).

Keywords

abstraction principlespredicativitysecond-order arithmeticFregeFrege’s Theorem
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 22 , Issue 1 , March 2016 , pp. 105 - 120
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

REFERENCES

Beth, Evert W., Chapter 13: Logicism, The Foundations of Mathematics: A Study in the Philosophy of Science, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1959, pp. 353–364.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. (editor), The Arché Papers on the Mathematics of Abstraction, The Western Ontario Series in Philosophy of Science, vol. 71, Springer, Berlin, 2007.Google Scholar
Cook, Roy T. (editor), Conservativeness, stability, and abstraction. British Journal for the Philosophy of Science, vol. 63 (2012), pp. 673–696.Google Scholar
Demopoulos, William (editor), Frege’s Philosophy of Mathematics, Harvard University Press, Cambridge, 1995.Google Scholar
Feferman, Solomon, Systems of predicative analysis. The Journal of Symbolic Logic, 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 $\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, Die Grundlagen der Arithmetik, Koebner, Breslau, 1884.Google Scholar
Frege, Gottlob, Grundgesetze der Arithmetik: begriffsschriftlich abgeleitet, Pohle, Jena, 1893, 1903, Two volumes. Reprinted in [12].Google Scholar
Frege, Gottlob, Grundgesetze der Arithmetik: begriffsschriftlich abgeleitet, Olms, Hildesheim, 1962.Google Scholar
Frege, Gottlob, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, second ed., Northwestern University Press, Evanston, 1980. Translated by Austin, John Langshaw.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
Ganea, Mihai, Burgess’ PV is Robinson’s Q. The Journal of Symbolic Logic, vol. 72 (2007), no. 2, pp. 618–624.Google Scholar
Hájek, Petr and Pudlák, Pavel, Metamathematics of First-Order Arithmetic, Perspectives in Mathematical Logic, Springer, Berlin, 1998.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
Heck, Richard G. Jr., Frege’s Theorem, Oxford University Press, Oxford, 2011.Google Scholar
Hodes, Harold, Logicism and the Ontological Commitments of Arithmetic. The Journal of Philosophy, vol. 81 (1984), no. 3, pp. 123–149.Google Scholar
Lindström, Per, Aspects of Incompleteness, second ed., Lecture Notes in Logic, vol. 10, Association for Symbolic Logic, Urbana, IL, 2003.Google Scholar
Linnebo, Øystein, Some criteria for acceptable abstraction. Notre Dame Journal of Formal Logic, vol. 52 (2010), no. 3, pp. 331–338.Google Scholar
Marker, David, Model Theory: An Introduction, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002.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 [6].Google Scholar
Simpson, Stephen G., Subsystems of Second Order Arithmetic, second ed., Cambridge University Press, Cambridge, 2009.Google Scholar
Visser, Albert, Categories of theories and interpretations, Logic in Tehran (Enayat, Ali, Kalantari, Iraj, and Moniri, Mojtaba, editors), Lecture Notes in Logic, vol. 26, Association for Symbolic Logic, La Jolla, 2006, pp. 284–341.CrossRefGoogle Scholar
Visser, Albert, The predicative Frege hierarchy. Annals of Pure and Applied Logic, vol. 160 (2009), no. 2, pp. 129–153.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, Logicism, interpretability, and knowledge of arithmetic. The Review of Symbolic Logic, vol. 7 (2014), no. 1, pp. 84–119.Google Scholar
Walsh, Sean, Fragments of Frege’s Grundgesetze and Gödel’s constructible universe. The Journal of Symbolic Logic, http://arxiv.org/abs/1407.3861, forthcoming.Google Scholar
Walsh, Sean, Predicativity, the Russell-Myhill paradox, and Church’s intensional logic. The Journal of Philosophical Logic, http://dx.doi.org/10.1007/s10992-015-9375-5, http://arxiv.org/abs/1506.02206, forthcoming.Google Scholar
Weyl, Hermann, Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis, Veit, Leipzig, 1918.Google Scholar
Wright, Crispin, Frege’s Conception of Numbers as Objects, Scots Philosophical Monographs, vol. 2, Aberdeen University Press, Aberdeen, 1983.Google Scholar