Linked bibliography for the SEP article "Set Theory: Constructive and Intuitionistic ZF" by Laura Crosilla

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  • Aczel, P., 1978, “The Type Theoretic Interpretation of Constructive Set Theory”, in Logic Colloquium ‘77, A. MacIntyre, L. Pacholski, J. Paris (eds.), Amsterdam and New York: North-Holland, pp. 55–66. (Scholar)
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  • –––, 1986, “The type theoretic interpretation of constructive set theory: inductive definitions”, in Logic, Methodology, and Philosophy of Science VII, R.B. Marcus, G.J. Dorn, and G.J.W. Dorn (eds.), Amsterdam and New York: North-Holland, pp. 17–49. (Scholar)
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  • Aczel, P., and Rathjen, M., 2001, “Notes on Constructive Set Theory”, Report No. 40, 2000/2001, Djursholm: Institut Mittag-Leffler, [available online] (Scholar)
  • Aczel, P., and Gambino, N., 2002, “Collection principles in dependent type theory”, in Types for Proofs and Programs (Lecture Notes in Computer Science 2277), P. Callaghan, Z. Luo, J. McKinna, and R. Pollack (eds.), Berlin: Springer, pp. 1–23. (Scholar)
  • Awodey, S., 2008, “A Brief Introduction to Algebraic Set Theory”, The Bulletin of Symbolic, 14 (3): 281–298. (Scholar)
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  • Beeson, M., 1985, Foundations of Constructive Mathematics, Berlin: Springer. (Scholar)
  • Bezem, M., Thierry, C. and Huber, S., 2014, “A model of type theory in cubical sets”, in 19th International Conference on Types for Proofs and Programs (TYPES 2013), Matthes, R. and Schubert, A. (eds.), Dagstuhl: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 107–128.
  • Bishop, E., 1967, Foundations of Constructive Analysis, New York: McGraw-Hill. (Scholar)
  • –––, 1970, “Mathematics as a numerical language”, in Intuitionism and Proof Theory, A. Kino, J. Myhill, and R. E. Vesley (eds.), Amsterdam: North-Holland, pp. 53–71. (Scholar)
  • Bishop, E., and Bridges, D., 1985, Constructive Analysis, Berlin and Heidelberg: Springer. (Scholar)
  • Bridges, D., and Richman, F., 1987, Varieties of Constructive Mathematics, Cambridge: Cambridge University Press. (Scholar)
  • Buchholz, W., Feferman, S., Pohlers, W., and Sieg, W., 1981, Iterated Inductive Definitions and Subsystems of Analysis, Berlin: Springer. (Scholar)
  • Cantini, A., and Crosilla, L., 2008, “Constructive set theory with operations”, in A. Andretta, K. Kearnes, D. Zambella (eds.), Logic Colloquium 2004 (Lecture Notes in Logic 29), Cambridge: Cambridge University Press, pp. 47–83. (Scholar)
  • Cantini, A., and Crosilla, L., 2010, “Explicit operational set theory”, in R. Schindler (ed.), Ways of Proof Theory, Frankfurt: Ontos, pp. 199–240. (Scholar)
  • Chen, R.-M. and Rathjen, M., 2012, “Lifschitz Realizability for Intuitionistic Zermelo-Fraenkel Set Theory”, Archive for Mathematical Logic, 51: 789–818. (Scholar)
  • Crosilla, L., 2017, “Predicativity and Feferman”, in G. Jäger and W. Sieg (eds.), Feferman on Foundations (Outstanding Contributions to Logic: Volume 13), Cham: Springer, pp 423–447. (Scholar)
  • Crosilla, L., and Rathjen, M., 2001, “Inaccessible set axioms may have little consistency strength”, Annals of Pure and Applied Logic, 115: 33–70. (Scholar)
  • Diaconescu, R., 1975, “Axiom of choice and complementation”, Proceedings of the American Mathematical Society, 51: 176–178. (Scholar)
  • Diener, H., and Lubarsky, R., 2013, “Separating the Fan Theorem and Its Weakenings”, in S. N. Artemov and A. Nerode (eds.), Proceedings of LFCS ‘13 (Lecture Notes in Computer Science 7734), Dordrecht: Springer, pp. 280–295. (Scholar)
  • Dummett, M., 2000, Elements of Intuitionism, second edition, (Oxford Logic Guides 39), New York: Oxford University Press. (Scholar)
  • Feferman, S., 1964, “Systems of predicative analysis”, Journal of Symbolic Logic, 29: 1–30. (Scholar)
  • –––, 1975, “A language and axioms for explicit mathematics”, in Algebra and Logic (Lecture Notes in Mathematics 450), J. Crossley (ed.), Berlin: Springer, pp. 87–139.
  • –––, 1988, “Weyl vindicated: Das Kontinuum seventy years later”, in Temi e prospettive della logica e della scienza contemporanee, C. Cellucci and G. Sambin (eds), pp. 59–93. (Scholar)
  • –––, 1993, “What rests on what? The proof-theoretic analysis of mathematics”, in Philosophy of Mathematics, Part I, Proceedings of the 15th International Wittgenstein Symposium. Vienna: Verlag Hölder-Pichler-Tempsky. (Scholar)
  • –––, 2005, “Predicativity”, in Handbook of the Philosophy of Mathematics and Logic, S. Shapiro (ed.), Oxford: Oxford University Press. (Scholar)
  • Fletcher, P., 2007, “Infinity”, in Handbook of the Philosophy of Logic, D. Jacquette, (ed.), Amsterdam: Elsevier, pp. 523–585. (Scholar)
  • Fourman, M.P., 1980, “Sheaf models for set theory”, Journal of Pure Applied Algebra, 19: 91–101. (Scholar)
  • Fourman, M.P., and Scott, D.S., 1980, “Sheaves and logic”, in Applications of Sheaves (Lecture Notes in Mathematics 753), M.P. Fourman, C.J. Mulvey and D.S. Scott (eds.), Berlin: Springer, pp. 302–401. (Scholar)
  • Friedman, H., 1973, “Some applications of Kleene’s methods for intuitionistic systems”, in Proceedings of the 1971 Cambridge Summer School in Mathematical Logic (Lecture Notes in Mathematics 337), A.R.D. Mathias and H. Rogers (eds.), Berlin: Springer, pp. 113–170. (Scholar)
  • –––, 1973a, “The consistency of classical set theory relative to a set theory with intuitionistic logic”, Journal of Symbolic Logic, 38: 315–319. (Scholar)
  • –––, 1977, “Set-theoretical foundations for constructive analysis”, Annals of Mathematics, 105: 1–28. (Scholar)
  • Friedman, H., Scedrov, A., 1983, “Set existence property for intuitionistic theories with dependent choice”, Annals of Pure and Applied Logic, 25: 129–140. (Scholar)
  • –––, 1984, “Large sets in intuitionistic set theory”, Annals of Pure and Applied Logic, 27: 1–24. (Scholar)
  • –––, 1985, “The lack of definable witnesses and provably recursive functions in intuitionistic set theory”, Advances in Mathematics, 57: 1–13. (Scholar)
  • Gambino, N., 2006, “Heyting-valued interpretations for constructive set theory”, Annals of Pure and Applied Logic, 137: 164–188. (Scholar)
  • Goodman, N.D., and Myhill, J., 1972, “The formalization of Bishop’s constructive mathematics”, in Toposes, Algebraic Geometry and Logic (Lecture Notes in Mathematics 274), F.W. Lawvere (ed.), Berlin: Springer, pp. 83–96. (Scholar)
  • Goodman, N.D., and Myhill, J., 1978, “Choice implies excluded middle”, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 24(5): 461. (Scholar)
  • Grayson, R.J., 1979, “Heyting-valued models for intuitionistic set theory”, in Applications of Sheaves (Lecture Notes in Mathematics 753), M.P. Fourman, C.J. Mulvey, and D.S. Scott (eds.), Berlin: Springer, pp. 402–414. (Scholar)
  • Griffor, E., and Rathjen, M., 1994, “The strength of some Martin-Löf type theories”, Archive Mathematical Logic, 33: 347–385. (Scholar)
  • van Heijenoort, J., 1967, From Frege to Gödel. A Source Book in Mathematical Logic 1879–1931, Cambridge: Harvard Univ. Press. (Scholar)
  • Kleene, S.C., 1945, “On the interpretation of intuitionistic number theory”, Journal of Symbolic Logic, 10: 109–124. (Scholar)
  • –––, 1962, “Disjunction and existence under implication in elementary intuitionistic formalisms”, Journal of Symbolic Logic, 27: 11–18. (Scholar)
  • –––, 1963, “An addendum”, Journal of Symbolic Logic, 28: 154–156. (Scholar)
  • Kreisel, G., 1958, “Ordinal logics and the characterization of informal concepts of proof”, Proceedings of the International Congress of Mathematicians (14–21 August 1958), Paris: Gauthier-Villars, pp. 289–299. (Scholar)
  • Kreisel, G., and Troelstra, A., S., 1970, “Formal systems for some branches of intuitionistic analysis”, Annals of Mathematical Logic, 1: 229–387. (Scholar)
  • Lifschitz, V., 1979, “CT\(_0\) is stronger than CT\(_0\)!”, Proceedings of the American Mathematical Society, 73(1): 101–106.
  • Lindström, I., 1989, “A construction of non-well-founded sets within Martin-Löf type theory”, Journal of Symbolic Logic, 54: 57–64. (Scholar)
  • Lipton, J., 1995, “Realizability, set theory and term extraction”, in The Curry-Howard isomorphism (Cahiers du Centre de Logique de l’Universite Catholique de Louvain 8), Louvain-la-Neuve: Academia, pp. 257–364. (Scholar)
  • Lorenzen, P., and Myhill, J., 1959, “Constructive definition of certain analytic sets of numbers”, Journal of Symbolic Logic, 24: 37–49. (Scholar)
  • Lubarsky, R., 2005, “Independence results around constructive ZF”, Annals of Pure and Applied Logic, 132: 209–225. (Scholar)
  • –––, 2006, “CZF and second order arithmetic”, Annals of Pure and Applied Logic, 141: 29–34.
  • –––, 2009, “Topological Forcing Semantics with Settling”, in S. N. Artemov and A. Nerode (eds.), Proceedings of LFCS ‘09 (Lecture Notes in Computer Science 5407), Dordrecht: Springer, pp. 309–322. (Scholar)
  • Lubarsky, R., and Rathjen, M., 2007, “On the Constructive Dedekind Reals”, in in S. N. Artemov and A. Nerode (eds.), Proceedings of LFCS 2007 (Lecture Notes in Computer Science 4514), Dordrecht: Springer, pp. 349–362. (Scholar)
  • MacLane, S., and Moerdijk, I., 1992, “Sheaves in Geometry and Logic”, New York: Springer. (Scholar)
  • Maietti, M.E., Sambin, G., 2005, “Toward a Minimalist Foundation for Constructive Mathematics”, in From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics (Oxford Logic Guides 48), L. Crosilla, and P. Schuster (eds.), Oxford: Oxford University Press. (Scholar)
  • Maietti, M.E., 2009, “A minimalist two-level foundation for constructive mathematics “, Annals of Pure and Applied Logic, 160(3): 319–354.
  • Martin-Löf, P., 1975, “An intuitionistic theory of types: predicative part”, in H.E. Rose and J. Sheperdson (eds.), Logic Colloquium ‘73, Amsterdam: North-Holland, pp. 73–118. (Scholar)
  • –––, 1984, “Intuitionistic Type Theory”, Naples: Bibliopolis. (Scholar)
  • McCarty, D.C., 1984, “Realisability and Recursive Mathematics”, D.Phil. Dissertation, Philosophy, Oxford University. (Scholar)
  • –––, 1986, “Realizability and recursive set theory”, Annals of Pure and Applied Logic, 32: 153–183. (Scholar)
  • Myhill, J., 1973, “Some properties of Intuitionistic Zermelo-Fraenkel set theory”, in Proceedings of the 1971 Cambridge Summer School in Mathematical Logic (Lecture Notes in Mathematics 337), A.R.D. Mathias, and H. Rogers(eds.), Berlin: Springer, pp. 206–231. (Scholar)
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  • Powell, W., 1975, “Extending Gödel’s negative interpretation to ZF”, Journal of Symbolic Logic, 40: 221–229. (Scholar)
  • Rathjen, M., Griffor, E., and Palmgren, E., 1998, “Inaccessibility in constructive set theory and type theory”, Annals of Pure and Applied Logic, 94: 181–200. (Scholar)
  • Rathjen, M., 1999, “The realm of ordinal analysis”, in Sets and Proofs (London Mathematical Society Lecture Notes 258), Cambridge: Cambridge University Press, pp. 219–279. (Scholar)
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  • –––, 2004, “Predicativity, circularity, and anti-foundation”, in One hundred years of Russell’s paradox (Logic and its Applications 6), G. Link (ed.), Berlin: de Gruyter, pp. 191–219. (Scholar)
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  • –––, 2012, “From the weak to the strong existence property”, Annals of Pure and Applied Logic, 163: 1400–1418. (Scholar)
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