Constructivist Versus Structuralist Foundations

• Aczel, P. 1978. The type theoretic interpretation of constructive set theory. Logic Colloquium ’77 (Proc. Conf., Wroclaw, 1977). Studies in logic and the foundations of mathematics, vol. 96, 55–66. Amsterdam/New York: North-Holland.

  Google Scholar 

• Awodey, S., and M.A. Warren. 2005/2006. Predicative algebraic set theory. Theory and Applications of Categories 15(1): 1–39.

  Google Scholar 

• Bishop, E. 1967. Foundations of constructive analysis. New York: McGraw-Hill.

  Google Scholar 

• Bishop, E. 1970a. Mathematics as a numerical language. In Intuitionism and proof theory, 53–71. Amsterdam: North-Holland.

  Google Scholar 

• Hofmann, M. 1994. On the interpretation of type theory in locally cartesian closed categories. In Proceedings of the Computer Science Logic ’94, Kazimierz, Poland, Lecture notes in computer science, vol. 933, ed. J. Tiuryn and L Pacholski. New York: Springer.

  Google Scholar 

• Johnstone, P.T. 2002. Sketches of an elephant: a Topos theory compendium, vols. 1, 2. New York: Oxford University Press.

  Google Scholar 

• Joyal, A., and I. Moerdijk. 1995. Algebraic set theory. London: Cambridge University Press.

  Book  Google Scholar 

• Lawvere, F.W. 2005. An elementary theory of the category of sets (long version). Theory and Applications of Categories 11: 7–35.

  Google Scholar 

• Lawvere, F.W., and R. Rosebrugh. 2003. Sets for mathematics. New York: Cambridge University Press.

  Book  Google Scholar 

• Mac Lane, S. 1998. Categories for the working mathematician, 2nd ed. New York: Springer.

  Google Scholar 

• Maietti, M.E. 2005. Modular correspondence between dependent type theories and categories including pretopoi and topoi. Mathematical Structures for Computer Science 15(6): 1089–1149.

  Article  Google Scholar 

• Maietti, M.E., and G. Sambin. 2005. Toward a minimalist foundation for constructive mathematics. In From sets and types to topology and analysis, Oxford Logic Guides, vol. 48, ed. L. Crosilla and P. Schuster, 91–114. Oxford: Oxford University Press.

  Google Scholar 

• Makkai, M. 1996. Avoiding the axiom of choice in general category theory. Journal of Pure and Applied Algebra 108: 109–173.

  Article  Google Scholar 

• Martin-Löf, P. 1975. An intuitionistic theory of types: predicative part. In Logic Colloquium ’73, ed. H.E. Rose and J. Shepherdson. Amsterdam: North-Holland.

  Google Scholar 

• McLarty, C. 2004. Exploring categorical structuralism. Philosophy of Mathematics 12: 37–53.

  Article  Google Scholar 

• McLarty, C. 2005. ETCS and philosophy of mathematics. Theory and Applications of Categories 11: 2–4.

  Google Scholar 

• Moerdijk, I., and E. Palmgren. 2000. Well-founded trees in categories. Annals of Pure and Applied Logic 104: 189–218.

  Article  Google Scholar 

• Moerdijk, I., and E. Palmgren. 2002. Type theories, toposes and constructive set theory: predicative aspects of AST. Annals of Pure and Applied Logic 114: 155–201.

  Article  Google Scholar 

• Myhill, J. 1975. Constructive set theory. Journal of Symbolic Logic 40(3): 347–382.

  Article  Google Scholar 

• Osius, G. 1974. Categorical set theory: a characterization of the category of sets. Journal of Pure and Applied Algebra 4: 79–119.

  Article  Google Scholar 

• Palmgren, E. Constructivist and structuralist foundations: Bishop’s and Lawvere’s theories of sets. Submitted to annals of pure and applied logic.

  Google Scholar 

• Tait, W.W. 2000. Cantor’s Grundlagen and the paradoxes of set theory. In Between logic and intuition: essays in honor of Charles Parsons, ed. G. Sher and R. Tieszen, 269–290. Cambridge: Cambridge University Press.

  Chapter  Google Scholar 

• van den Berg, B. 2005. Inductive types and exact completions. Annals of Pure and Applied Logic 134: 95–121.

  Article  Google Scholar 

• van den Berg, B., and I. Moerdijk. 2008. Aspects of predicative algebraic set theory I: exact completion. Annals of Pure and Applied Logic 156: 123–159.

  Article  Google Scholar 

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