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Published online by Cambridge University Press:  10 February 2015


The big question at the end of Feferman (2013) is: Is it possible to find a foundation for unlimited category theory? I show that the answer is no by showing that unlimited category theory is inconsistent.

Research Article
Copyright © Association for Symbolic Logic 2015 

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Brown, R., Morris, I., Shrimpton, J., & Wensley, C. D. (2008). Graphs of morphisms of graphs. Electronic Journal of Combinatorics, 15(1), Article 1, 28.Google Scholar
Bumby, R. T., & Latch, D. M. (1986). Categorical constructions in graph theory. International Journal of Mathematics and Mathematical Sciences, 9(1), 116.CrossRefGoogle Scholar
Delhommé, C., & Morillon, M. (2006). Spanning graphs and the axiom of choice. Reports on Mathematical Logic, 40, 165180.Google Scholar
Eilenberg, S., & Mac Lane, S. (1945). General theory of natural equivalences. Transactions of the American Mathematical Society, 58, 231294.CrossRefGoogle Scholar
Feferman, S. (1969). Set-theoretical foundations of category theory. In Mac Lane, S., editor, Reports of the Midwest Category Seminar III. Lecture Notes in Mathematics, Vol. 106, Berlin: Springer, pp. 201247.CrossRefGoogle Scholar
Feferman, S. (2013). Foundations of unlimited category theory: What remains to be done. The Review of Symbolic Logic, 6(1), 615.CrossRefGoogle Scholar
Harary, F. (1969). Graph Theory. Reading: Addison-Wesley.CrossRefGoogle Scholar
Krömer, R. (2007). Tool and Object. Basel: Birkhäuser Verlag.Google Scholar
Lawvere, F. (1969). Diagonal arguments and cartesian closed categories. In Hilton, P. J., editor, Category Theory, Homology Theory and Their Applications II. Lecture Notes in Mathematics, Vol. 92, Berlin: Springer-Verlag, pp. 134145.CrossRefGoogle 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(6), 1506.CrossRefGoogle ScholarPubMed
Lawvere, F. W. (1966). The category of categories as a foundation for mathematics. In Eilenberg, S., Harrison, D. K., Röhrl, H., Mac Lane, S., editors, Proceedings of the La jolla conference on categorical algebra, La Jolla. New York: Springer-Verlag, pp. 121.Google Scholar
Mac Lane, S. (1969). One universe as a foundation for category theory. In Mac Lane, S., editor, Reports of the Midwest Category Seminar III. Lecture Notes in Mathematics, Vol. 106, Berlin: Springer, pp. 192200.CrossRefGoogle Scholar
Mac Lane, S. (1971). Categories for the Working Mathematician. New York: Springer-Verlag.CrossRefGoogle Scholar
Marquis, J. P. (2009). From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory. Dordrecht: Springer.Google Scholar
McLarty, C. (1991). Axiomatizing a category of categories. Journal of Symbolic Logic, 56, 12431260.CrossRefGoogle Scholar
McLarty, C. (1992). Failure of Cartesian closedness in NF. The Journal of Symbolic Logic, 57(2), 555556.CrossRefGoogle Scholar
McLarty, C. (1995). Elementary Categories, Elementary Toposes. Oxford: Clarendon Press.Google Scholar
Plessas, D. J. (2011). The categories of graphs. PhD Thesis, The University of Montana.Google Scholar
Scheinerman, E. A. (2012). Mathematics: A Discrete Introduction. Boston: Brooks/Cole Publishing Company.Google Scholar
Shulman, M. A. (2008). Set theory for category theory. arXiv preprint arXiv:0810.1279.
Zermelo, E. (1930). On boundary numbers and domains of sets. In Ewald, W., editor. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 1996. New York: Oxford University Press, pp. 12191233.Google Scholar