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FOUNDATIONS OF UNLIMITED CATEGORY THEORY: WHAT REMAINS TO BE DONE

Published online by Cambridge University Press:  01 June 2012

SOLOMON FEFERMAN
SOLOMON FEFERMAN*
Affiliation:
Department of Mathematics, Stanford University
*
*DEPARTMENT OF MATHEMATICS, STANFORD UNIVERSITY, STANFORD, CA 94305 E-mail: feferman@stanford.edu
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Abstract

Following a discussion of various forms of set-theoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unlimited” or “naive” category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had been met in that approach, and finally explains what remains to be done if one is to have a fully satisfactory solution.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 1 , March 2013 , pp. 6 - 15
Copyright
Copyright © Association for Symbolic Logic 2012

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References

BIBLIOGRAPHY

Ehrenfeucht, A., & Mostowski, A. (1956). Models of axiomatic theories admitting automorphisms. Fundamenta Mathematicae, 43, 50–68.Google Scholar
Eilenberg, S., & Mac Lane, S. (1945). General theory of natural equivalences. Transactions of the American Mathematical Society, 58, 231–294.Google Scholar
Engeler, E., & Röhrl, H. (1969). On the problem of foundations of category theory. Dialectica, 23, 58–66.Google Scholar
Feferman, S. (1969). Set-theoretical foundations for category theory (with an appendix by G. Kreisel). In Barr, M., et al. ., editors. Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics, Vol. 106. Berlin, Germany: Springer, pp. 201–247.Google Scholar
Feferman, S. (1974). Some formal systems for the unlimited theory of structures and categories. Unpublished MS, available online athttp://math.stanford.edu/ ∼ feferman/papers/Unlimited.pdf. Abstract in Journal of Symbolic Logic, 39(1974), 374–375.Google Scholar
Feferman, S. (1975). A language and axioms for explicit mathematics. In Algebra and Logic, Lecture Notes in Mathematics, Vol. 450. Berlin, Germany: Springer, pp. 87–139.CrossRefGoogle Scholar
Feferman, S. (1977). Categorical foundations and foundations of category theory. In Butts, R. E., and Hintikka, J., editors. Logic, Foundations of Mathematics and Computability Theory, Vol. 1. Dordrecht, The Netherlands: Reidel, pp. 149–165.Google Scholar
Feferman, S. (2004). Typical ambiguity: Trying to have your cake and eat it too. In Link, G., editor. One Hundred Years of Russell’s Paradox. Berlin: De Gruyter, pp. 135–151.CrossRefGoogle Scholar
Feferman, S. (2006). Enriched stratified systems for the foundations of category theory. In Sica, G., editor. What is Category Theory? Monza, Italy: Polimetrica. Reprinted in (G. Sommaruga, ed.) Foundational Theories of Classical and Constructive Mathematics (2011). Berlin: Springer, pp. 127–143.Google Scholar
Feferman, S. (2009). Operational set theory and small large cardinals. Information and Computation, 207, 971–979.Google Scholar
Freyd, P. (1964). Abelian Categories. New York: Harper & Row.Google Scholar
Hellman, G. (2003). Does category theory provide a framework for mathematical structuralism?. Philosophia Mathematica III, 11, 129–157.CrossRefGoogle Scholar
Jensen, R. (1969). On the consistency of a slight (?) modification of Quine’s New Foundations. In Davidson, D., and Hintikka, J., editors. Words and Objections. Essays on the work of W.V.O. Quine. Dordrecht, The Netherlands: Reidel, pp. 278–291.Google Scholar
Kock, A. (1991). Synthetic Differential Geometry. Cambridge, UK: Cambridge University Press.Google Scholar
Lawvere, F. W. (1964). An elementary theory of the category of sets. Proceedings of the National Academy of Sciences of the United States of America, 52, 1506–1511.Google Scholar
Lawvere, F. W. (1966). The category of all categories as a foundation for mathematics. In Proceedings of the La Jolla Conference on Categorical Algebra. Berlin: Springer-Verlag, pp. 1–20.Google Scholar
Linnebo, ø., & Pettigrew, R. (2011). Category theory as an autonomous foundation. Philosophia Mathematica III, 19, 227–254.Google Scholar
Mac Lane, S. (1961). Locally small categories and the foundations of mathematics. In Infinitistic Methods. Oxford: Pergamon Press, pp. 25–43.Google Scholar
Mac Lane, S. (1969). One universe as a foundation for category theory. In Barr, M., et al. ., editors. Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics, Vol. 106. Berlin, Germany: Springer, pp. 192–200.Google Scholar
Mac Lane, S. (1971). Categories for the Working Mathematician. Berlin: Springer-Verlag. Second edition, 1998.CrossRefGoogle Scholar
Mac Lane, S. (1986). Mathematics: Form and Function. Berlin: Springer-Verlag.Google Scholar
McLarty, C. (1991). Axiomatizing a category of categories, Journal of Symbolic Logic, 56, 1243–1260.CrossRefGoogle Scholar
Oberschelp, A. (1973). Set theory over classes, Dissertationes Mathematicae (Rozprawy Matematyczne), 106, 1–62.Google Scholar
Osius, G. (1974). Categorical set theory: A characterization of the category of sets. Journal of Pure and Applied Algebra, 4, 79–119.Google Scholar
Quine, W. V. O. (1937). New foundations for mathematical logic. American Mathematical Monthly, 44, 77–80.Google Scholar
Shulman, M. (2008). Set theory for category theory. arXiv:0810.1279v2 [math.CT] 7 Oct 2008.Google Scholar
Specker, E. (1962). Typical ambiguity in logic, In Nagel, E., et al. ., editors. Methodology and Philosophy of Science. Proceedings of the 1960 International Congress. Stanford, CA: Stanford University Press, pp. 116–123.Google Scholar