Linked bibliography for the SEP article "The Axiom of Choice" by John L. Bell

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  • Aczel, P., 1978. “The type-theoretic interpretation of constructive set theory,” in A. ManIntyre, L. Pacholski, and J. Paris (eds.), Logic Colloquium 77, Amsterdam: North-Holland, pp. 55–66. (Scholar)
  • –––, 1982. “The type-theoretic interpretation of constructive set theory: choice principles,” in A. S. Troelstra and D. van Dalen (eds.), The L.E.J. Brouwer Centenary Symposium, Amsterdam: North-Holland, pp. 1–40. (Scholar)
  • Aczel, P. and N. Gambino, 2002. “Collection principles in dependent type theory,” in P. Callaghan, Z. Luo, J. McKinna and R. Pollack (eds.), Types for Proofs and Programs (Lecture Notes on Computer Science, Volume 2277), Berlin: Springer, pp. 1–23. (Scholar)
  • –––, 2005. “The generalized type-theoretic interpretation of constructive set theory,” Journal of Symbolic Logic, 71/1: 67–103. [Preprint available online in compressed Postscript] (Scholar)
  • Aczel, P., Berg, B. and J. Granstrom, 2013. “Are there enough injective sets?,” Studia Logica, 101(3): 467–482. (Scholar)
  • Aczel, P. and M. Rathjen, 2001. Notes on Constructive Set Theory. Technical Report 40, Mittag-Leffler Institute, The Swedish Royal Academy of Sciences. [Preprint available online] (Scholar)
  • Banach, S. and Tarski, A., 1924. “Sur la décomposition des ensembles de points en parties respectivement congruentes,” Fundamenta Mathematicae, 6: 244–277. (Scholar)
  • Bell, J.L., 1983. “On the strength of the Sikorski extension theorem for Boolean algebras,” Journal of Symbolic Logic, 48: 841–846. (Scholar)
  • –––, 1988. Toposes and Local Set Theories: An Introduction, Oxford: Clarendon Press. (Scholar)
  • –––, 1997. “Zorn’s lemma and complete Boolean algebras in intuitionistic type theories,” Journal of Symbolic Logic, 62: 1265–1279. (Scholar)
  • –––, 2003. “Some new intuitionistic equivalents of Zorn’s Lemma,” Archive for Mathematical Logic, 42: 811–814. (Scholar)
  • –––, 2005. Set Theory: Boolean-valued Models and Independence Proofs, Oxford: Clarendon Press. (Scholar)
  • –––, 2006. “Choice principles in intuitionistic set theory,” in A Logical Approach to Philosophy, Devidi, D. and Kenyon, T.(eds.), Berlin: Springer, 36–44. (Scholar)
  • –––, 2008. “The axiom of choice and the law of excluded middle in weak set theories,” Mathematical Logic Quarterly, 48: 841–846. (Scholar)
  • –––, 2009. The Axiom of Choice, London: College Publications. (Scholar)
  • –––, 2011. “The axiom of choice in an elementary theory of operations and sets” in Analysis and Interpretation in the Exact Sciences, Devidi, D. and Kenyon, T.(eds.), Berlin: Springer, 163–175. (Scholar)
  • Bell, J.L. and Fremlin, D., 1972. “The maximal ideal theorem for lattices of sets,” Bulletin of the London Mathematical Society, 4: 1–2. (Scholar)
  • –––, 1972a. “A geometric form of the axiom of choice,” Fundamenta Mathematicae, 77: 167–170.
  • Bell, J.L. and Machover, M. , 1977. A Course in Mathematical Logic. Amsterdam: North-Holland. (Scholar)
  • Bernays, P., 1942. “A system of axiomatic set theory, Part III,” Journal of Symbolic Logic, 7: 65–89. (Scholar)
  • Bishop, E. and Bridges, D., 1985. Constructive Analysis, Berlin: Springer. (Scholar)
  • Blass, A., 1984. “Existence of bases implies the axiom of choice,” in Axiomatic Set Theory, Baumgartner, Martin and Shelah (eds.) (Contemporary Mathematics Series, Volume 31), American Mathematical Society, pp. 31–33. (Scholar)
  • Bochner, S., 1928. “Fortsetzung Riemannscher Flachen,” Mathematische Annalen, 98: 406–421. (Scholar)
  • Bourbaki, N., 1939. Elements de Mathematique, Livre I: Theorie des Ensembles, Paris: Hermann. (Scholar)
  • –––, 1950.“Sur le theoreme de Zorn,” Archiv dem Mathematik, 2: 434–437. (Scholar)
  • Cohen, P.J., 1963. “The independence of the continuum hypothesis I,” Proceedings of the U.S. National Academy of Sciemces, 50: 1143–48. (Scholar)
  • –––, 1964. “The independence of the continuum hypothesis II,” Proceedings of the U.S. National Academy of Sciemces, 51: 105–110. (Scholar)
  • –––, 1966. Set Theory and the Continuum Hypotheis, New York: Benjamin. (Scholar)
  • Curry, H.B. and R. Feys, 1958. Combinatory Logic, Amsterdam: North Holland. (Scholar)
  • Devidi, D., 2004. “Choice principles and constructive logics,” Philosophia Mathematica, 12(3): 222–243. (Scholar)
  • Diaconescu, R., 1975. “Axiom of choice and complementation,” Proceedings of the American Mathematical Society, 51: 176–8. (Scholar)
  • Fraenkel, A., 1922. “Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre”, Mathematische Annalen, 86: 230–237. (Scholar)
  • Fraenkel, A., 1922a.“Über den Begriff ‘definit’ und die Unabhängigkeit des Auswahlsaxioms,” Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physik-math. Klasse, 253–257. Translated in van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Harvard University Press, 1967, pp. 284–289. (Scholar)
  • Fraenkel, A., Y. Bar-Hillel and A. Levy, 1973. Foundations of Set Theory, Amsterdam: North-Holland, 2nd edition. (Scholar)
  • Gödel, K., 1938. “The consistency of the axiom of choice and of the generalized continuum-hypothesis,” Proceedings of the U.S. National Academy of Sciences, 24: 556–7. (Scholar)
  • Gödel, K., 1938. “Consistency-proof for the generalized continuum-hypothesis,” Proceedings of the U.S. National Academy of Sciemces, 25: 220–4. (Scholar)
  • Gödel, K., 1940. The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory, Annals of Mathematics Studies, No. 3, Princeton: Princeton University Press. (Scholar)
  • Gödel, K., 1964. “Remarks before the Princeton Bicentennial Conference,” in The Undecidable, Martin Davis (ed.), CITY: Raven Press, pp. 84–88. (Scholar)
  • Goodman, N. and Myhill, J., 1978. “Choice implies excluded middle,” Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik , 24(25–30): 461. (Scholar)
  • Grattan-Guinness, I., 2012. “Jourdain, Russell and the axiom of choice: a new document,” Russell: The Journal of the Bertrand Russell Archives, 32(1): 69–74. (Scholar)
  • Grayson, R.J., 1975. “A sheaf approach to models of set theory,” M.Sc. Thesis, Mathematics Department, Oxford University.
  • Halpern, , J.D. and Levy, A., 1971. “The Boolean prime ideal theorem does not imply the axiom of choice,” Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, Vol. XIII, Part I. American Mathematical Society, pp. 83–134. (Scholar)
  • Hamel, G., 1905. “Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: f(x + y) = f(x) + f(y),” Mathematische Annalen, 60: 459–62. (Scholar)
  • Hausdorff, F., 1909. “Die Graduierung nach dem Endverlauf,” Königlich Sächsichsen Gesellschaft der Wissenschaften zu Leipzig, Math.—Phys. Klasse, Sitzungberichte, 61: 297–334. (Scholar)
  • –––, 1914. Grundzüge der Mengenlehre, Leipzig: de Gruyter. Reprinted, New York: Chelsea, 1965. (Scholar)
  • –––, 1914a. “Bemerkung über den Inhalt von Punktmengen,” Mathematische Annalen, 75: 428–433. (Scholar)
  • Hilbert D., 1926. “Über das Unendliche,” Mathematische Annalen, 95. Translated in J. van Heijenoort (ed.) From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967, pp. 367–392. (Scholar)
  • Hodges, W., 1979. “Krull implies Zorn,” Journal of the London Mathematical Society, 19: 285–7. (Scholar)
  • Howard, P. and Rubin, J. E., 1998. Consequences of the Axiom of Choice, American Mathematical Society Surveys and Monographs, Vol. 59. (Scholar)
  • Howard, W. A., 1980. “The formulae-as-types notion of construction,” in J. R. Hindley and J. P. Seldin (eds.), To H. B. Curry: Essays on Combinatorial Logic. Lambda Calculus and Formalism, New York and London: Academic Press, pp. 479–490. (Scholar)
  • Jacobs, B., 1999. Categorical Logic and Type Theory, Amsterdam: Elsevier. (Scholar)
  • Jech, T., 1973. The Axiom of Choice, Amsterdam: North-Holland. (Scholar)
  • Kelley, J.L., 1950. “The Tychonoff product theorem implies the axiom of choice,” Fundamenta Mathematicae, 37: 75–76. (Scholar)
  • Klimovsky, G., 1958. “El teorema de Zorn y la existencia de filtros a ideales maximales en los reticulados distributivos,” Revista de la Union Matematica Argentina , 18: 160–64. (Scholar)
  • Kuratowski, K., 1922. “Une méthode d’élimination des nombres transfinis des raissonements mathématiques,” Fundamenta Mathematicae, 3: 76–108. (Scholar)
  • Lawvere, F. W. and Rosebrugh, R., 2003. Sets for Mathematics, Cambridge: Cambridge University Press. (Scholar)
  • Lindenbaum, A., and Mostowski, A., 1938. “Über die Unabhängigkeit des Auswahlsaxioms und einiger seiner Folgerungen,” Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, 31: 27–32. (Scholar)
  • Maietti, M.E., 2005. “Modular correspondence between dependent type theories and categories including pretopoi and topoi,” Mathematical Structures in Computer Science, 15/6: 1089–1145. (Scholar)
  • Martin-Löf, P., 1975. “An Intuitionistic theory of types; predicative part,” in H. E. Rose and J. C. Shepherdson (eds.), Logic Colloquium 73, Amsterdam: North-Holland, pp. 73–118. (Scholar)
  • –––, 1982. “Constructive mathematics and computer programming,” in L. C. Cohen, J. Los, H. Pfeiffer, and K.P. Podewski (eds.), Logic, Methodology and Philosophy of Science VI, Amsterdam: North-Holland, pp. 153–179. (Scholar)
  • –––, 1984. Intuitionistic Type Theory, Naples: Bibliopolis. (Scholar)
  • –––, 2006. “100 years of Zermelo’s axiom of choice: what was the problem with it?,” The Computer Journal, 49(3): 345–350. (Scholar)
  • Mendelson, E., 1956. “The independence of a weak axiom of choice,” Journal of Symbolic Logic, 21: 350–366. (Scholar)
  • –––, 1958. “The axiom of fundierung and the axiom of choice,” Arkiv fur Mathematische Logik und Grundlagenforschung, 4: 67–70. (Scholar)
  • –––, 1987. Introduction to Mathematical Logic, CITY: Wadsworth & Brooks, 3rd edition. (Scholar)
  • Moore, G.H., 1982. Zermelo’s Axiom of Choice, Berlin: Springer-Verlag. (Scholar)
  • Moore, R.L., 1932. Foundations of Point Set Theory, Anerican Mathematical Society Colloquium Publications, vol. 13. (Scholar)
  • Myhill, J. and Scott, D.S., 1971. “Ordinal definability,” Axiomatic Set Theory. Proceedings of Symposia in Pure Mathematics, Vol. XIII, Part I. American Mathematical Society, pp. 271–8. (Scholar)
  • Post, E.L., 1953. “A necessary condition for definability for transfinite von Neumann-Gödel set theory sets, with an application to the problem of the existence of a definable well-ordering of the continuum.” Preliminary Report, Bulletin of the American Mathematical Society, 59: 246. (Scholar)
  • Ramsey, F. P., 1926. “The foundations of mathematics,” Proceedings of the London Mathematical Society, 25: 338–84. Reprinted in The Foundations of Mathematics and Other Essays, D.H. Mellor, ed. London: Routledge, 2001. (Scholar)
  • Rubin, H. and Rubin, J. E., 1985. Equivalents of the Axiom of Choice II, Amsterdam: North-Holland. (Scholar)
  • Rubin, H. and Scott, D.S., 1954. “Some topological theorems equivalent to the prime ideal theorem,” Bulletin of the American Mathematical Society, 60: 389. (Scholar)
  • Russell, B., 1906. “On some difficulties in the theory of transfinite numbers and order types,” Proceedings of the London Mathematical Society, 4(2): 29–53. (Scholar)
  • Shoenfield, J. R., 1955. “The independence of the axiom of choice,” Journal of Symbolic Logic, 20: 202. (Scholar)
  • Sikorski, R., 1948. “A theorem on extensions of homomorphisms,” Annales de la Societé Polonaise de Mathématiques, 21: 332–35.
  • Solovay, R., 1970. “A model of set theory in which every set of reals is Lebesgue measurable,” Annals of Mathematics, 92: 1–56. (Scholar)
  • Specker, E., 1957. “Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom),” Zeit. Math. Logik und Grund., 3: 173–210. (Scholar)
  • Steinitz, E., 1910. “Algebraische Theorie der Körper,” Journal für die Reine und angewandte Mathematik (Crelle), 137: 167–309. (Scholar)
  • Stone, M.H., 1936. “The theory of representations for Boolean algebras,” Transactions of the American Mathematical Society, 40: 37–111. (Scholar)
  • Tait, W. W., 1994. “The law of excluded middle and the axiom of choice,” in Mathematics and Mind, A. George (ed.), New York: Oxford University Press, pp. 45–70. (Scholar)
  • Takeuti, G., 1961. “Remarks on Cantor’s Absolute,” Journal of the Mathematical Society of Japan, 13: 197–206. (Scholar)
  • Tarski, A., 1948. “Axiomatic and algebraic aspects of two theorems on sums of cardinals,” Fundamenta Mathematicae, 35: 79–104. (Scholar)
  • Teichmuller, O., 1939. “Brauch der Algebraiker das Auswahlaxiom?Deutsches Mathematik, 4: 567–577. (Scholar)
  • Vitali, G., 1905. Sul problema della misura dei gruppi di punti di una retta, Bologna: Tip. Gamberini e Parmeggiani. (Scholar)
  • Wagon, S., 1993.The Banach-Tarski Paradox, Cambridge University Press. (Scholar)
  • Zermelo, E., 1904. “Neuer Beweis, dass jede Menge Wohlordnung werden kann (Aus einem an Herrn Hilbert gerichteten Briefe)”, Mathematische Annalen, 59: 514–16. Translated in J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967, pp. 139–141. (Scholar)
  • –––, 1908. Neuer Beweis für die Möglichkeit einer Wohlordnung, Mathematische Annalen, 65: 107–128. Translated in J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967, pp. 183–198. (Scholar)
  • –––, 1908a.“Untersuchungen uber die Grundlagen der Mengenlehre,” Mathematische Annalen, 65: 107–128. Translated in J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967, pp. 199–215. (Scholar)
  • Zorn, M., 1935. A remark on method in transfinite algebra, Bulletin of the American Mathematical Society, 41: 667–70. (Scholar)

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