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The purpose of this paper is to investigate categoricity arguments conducted in second order logic and the philosophical conclusions that can be drawn from them. We provide a way of seeing this result, so to speak, through a first order lens divested of its second order garb. Our purpose is to draw into sharper relief exactly what is involved in this kind of categoricity proof and to highlight the fact that we should be reserved before drawing powerful philosophical conclusions from it.

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Chang C. C., & Keisler H. J. (1973). Model Theory. Amsterdam, the Netherlands: North Holland Publishing Company.
Church A. (1940). A formulation of the simple theory of types. Journal of Symbolic Logic, 5 (2), 5568.
Dedekind R. (1963). Essays on the Theory of Numbers. New York, NY: Dover Publications.
Feferman S. (1964). Systems of predicative analysis. Journal of Symbolic Logic, 29 (1), 130.
Feferman S. (1999). Does mathematics need new axioms? American Mathematical Monthly, 6, 401446.
Gitman V., & Hamkins J. D. (2010). A natural model of the multiverse axioms. Notre Dame Journal of Formal Logic, 51 (4), 475484.
Halbach V., & Horsten L. (2005). Computational structuralism. Philosophia Mathematica, 13, 174186.
Hamkins J. (2007). The modal logic of forcing. Transactions of the Americal Mathematical Society, 360 (4), 17931817.
Hamkins J. (2009). Some second order set theory. In Ramanujan R., and Sarukkai S., editors, Logic and Its Applications: Lecture Notes in Computer Science, Vol. 5378. Heidelberg, Germany: Springer-Verlag, pp. 3650.
Hamkins J. (2012). The set-theoretic multiverse. The Review of Symbolic Logic, 5, 416449.
Kaye R. (1991). Models of Peano Arithmetic. Oxford, UK: Oxford University Press.
Kreisel G. (1969). Informal rigour and completeness proofs. In Hintikka J., editor. The Philosophy of Mathematics. London, UK: Oxford University Press.
Marker D. (2002). Model Theory and Introduction. New York, NY: Springer.
Martin D. A. (2001). Multiple universes of sets and indeterminate truth values. Topoi, 20 (1), 516.
Mayberry J. P. (2000). Review of J. L. Bell: A primer of infinitesimal analysis. British Journal of the Philosophy of Science, 51, 339445.
McGee V. (1997). How we learn mathematical language. The Philosophical Review, 106 (1), 3568.
Mostowski A. (1967). Recent results in set theory. In Lakatos I., editor. Problems in the Philosophy of Mathematics. Amsterdam, The Netherlands: North Holland Publishing Company.
Read S. (1997). Completeness and categoricity: Frege, Gödel and model theory. History and Philosophy of Logic, 18 (2), 7993.
Shapiro S. (1985). Second-order languages and mathematical practices. The Journal of Symbolic Logic, 50, 714742.
Shapiro S. (1990). Second-order logic foundations and rules. The Journal of Philosophy, 87 (5), 234261.
Shapiro S. (1991). Foundations without Foundationalism: A Case for Second Order Logic. Oxford, UK: Oxford University Press.
Shapiro S. (1997). Philosophy of Mathematics: Structure and Ontology. Oxford, UK: Oxford University Press.
Shepherdson J. C. (1952). Inner models for set theory—Part II. Journal of Symbolic Logic, 17 (4), 225237.
Simpson S. G. (1999). Subsystems of Second Order Arithmetic. Berlin, Germany: Springer.
Simpson S., & Yokoyama K. (2012). On the reverse mathematics of Peano categoricity. Unpublished manuscript.
Väänänen J. (2012). Second order logic or set theory? Bulletin of Symbolic Logic, 18, 91121.
Weston T. (1976). Kreisel, the continuum hypothesis and second order set theory. The Journal of Philosophical Logic, 5, 281298.
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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
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