Linked bibliography for the SEP article "Alternative Axiomatic Set Theories" by M. Randall Holmes

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  • Aczel, Peter, 1978, “The Type Theoretic Interpretation of Constructive Set Theory”, in A. MacIntyre, L. Pacholski, J. Paris (eds.), Logic Colloquium ‘77, (Studies in Logic and the Foundations of Mathematics, 96), Amsterdam: North-Holland, pp. 55–66. doi:10.1016/s0049-237x(08)71989-x (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, (Studies in Logic and the Foundations of Mathematics, 110), Amsterdam: North-Holland, pp. 1–40. doi:10.1016/s0049-237x(09)70120-x (Scholar)
  • –––, 1986, “The Type Theoretic Interpretation of Constructive Set Theory: Inductive Definitions”, in Ruth Barcan Marcus, Georg J.W.Dorn, and Paul Weingartner (eds.), Logic, Methodology, and Philosophy of Science VII, (Studies in Logic and the Foundations of Mathematics, 114), Amsterdam: North-Holland, pp. 17–49. doi:10.1016/s0049-237x(09)70683-4 (Scholar)
  • –––, 1988, Non-Well-Founded Sets (CSLI Lecture Notes, 14), Stanford: CSLI Publications. (Scholar)
  • St. Augustine, De Civitate Dei, Book 12, chapter 18.
  • Barwise, Jon, 1975, Admissible Sets and Structures: An Approach to Definability Theory, (Perspectives in Mathematical Logic, 7), Berlin: Springer-Verlag. (Scholar)
  • Boffa, M., 1988, “ZFJ and the Consistency Problem for NF”, Jahrbuch der Kurt Gödel Gesellschaft, Vienna, pp. 102–106 (Scholar)
  • Burali-Forti, C., 1897, “Una questione sui numeri transfiniti”, Rendiconti del Circolo matematico di Palermo, 11(1): 154–164. A correction appears in “Sulle classi ben ordinate”, Rendiconti del Circolo matematico di Palermo, 11(1): 260. It is not clear that Burali-Forti ever correctly understood his paradox. doi:10.1007/bf03015911 and doi:10.1007/BF03015919 (Scholar)
  • Cantor, Georg, 1872, “Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen”, Mathematischen Annalen, 5: 123–32. (Scholar)
  • –––, 1891, “Über eine elementare Frage der Mannigfaltigkeitslehre”, Jahresbericht der deutschen Mathematiker-Vereiningung, 1: 75–8. (Scholar)
  • Cocchiarella, Nino B., 1985, “Frege’s Double-Correlation Thesis and Quine’s Set Theories NF and ML”, Journal of Philosophical Logic, 14(1): 1–39. doi:10.1007/bf00542647 (Scholar)
  • Crabbé, Marcel, 1982, “On the Consistency of an Impredicative Subsystem of Quine’s NF”, Journal of Symbolic Logic, 47(1): 131–36. doi:10.2307/2273386 (Scholar)
  • –––, 2016, “NFSI is not included in NF3”, Journal of Symbolic Logic, 81(3): 948–950. doi:10.1017/jsl.2015.29
  • Dedekind, Richard, 1872, Stetigkeit und irrationale Zahlen, Brannschweig: Friedrich Vieweg und Sohn (second edition, 1892). (Scholar)
  • Esser, Olivier, 1999, “On the Consistency of a Positive Theory”, Mathematical Logic Quarterly, 45(1): 105–116. doi:10.1002/malq.19990450110 (Scholar)
  • Feferman, Sol, 2006, “Enriched Stratified Systems for the Foundations of Category Theory” in Giandomenico Sica (ed.), What is Category Theory?, Milan: Polimetrica. [Feferman 2006 preprint available online (PDF)] (Scholar)
  • Frege, Gottlob, 1884, Die Grundlagen der Arithmetik, English translation by J.L. Austin, The Foundations of Arithmetic, Oxford: Blackwell, 1974. (Scholar)
  • Friedman, Harvey, 1973, “Some Applications of Kleene’s Methods for Intuitionistic Systems”, in A.R.D. Mathias and H. Rogers (eds.), Cambridge Summer School in Mathematical Logic, (Lecture Notes in Mathematics, 337), Berlin: Springer-Verlag, pp. 113–170. doi:10.1007/bfb0066773 (Scholar)
  • Grishin, V.N., 1969, “Consistency of a Fragment of Quine’s NF System”, Soviet Mathematics Doklady, 10: 1387–1390. (Scholar)
  • Hallett, Michael, 1984, Cantorian Set Theory and Limitation of Size, Oxford: Clarendon, pp. 280–286. (Scholar)
  • Hamkins, Joel David, 2012, “The Set-Theoretic Multiverse”, Review of Symbolic Logic, 5(3): 416–449. doi:10.1017/s1755020311000359 (Scholar)
  • Holmes, M. Randall, 1998, Elementary Set Theory with a Universal Set, (Cahiers du Centre de logique, 10), Louvain-la-Neuve: Academia. (See chapter 20 for the discussion of well-founded extensional relation types.) [Holmes 1998 revised and corrected version available online (PDF)] (Scholar)
  • –––, 2012, “The Usual Model Construction for NFU Preserves Information”, Notre Dame Journal of Formal Logic, 53(4): 571–580. doi:10.1215/00294527-1722764 (Scholar)
  • Jensen, Ronald Bjorn, 1968, “On the Consistency of a Slight (?) Modification of Quine’s ‘New Foundations’”, Synthese, 19(1): 250–63. doi:10.1007/bf00568059 (Scholar)
  • Kisielewicz, Andrzej, 1998, “A Very Strong Set Theory?”, Studia Logica, 61(2): 171–178. doi:10.1023/a:1005048329677 (Scholar)
  • Kuratowski, Casimir [Kazimierz], 1921, “Sur la notion de l’ordre dans la Théorie des Ensembles”, Fundamenta Mathematicae, 2(1): 161–171. [Kuratowski 1921 available online] (Scholar)
  • Lévy, Azriel, 1959, “On Ackermann’s Set Theory”, Journal of Symbolic Logic, 24(2): 154–166. doi:10.2307/2964757 (Scholar)
  • Mac Lane, Saunders, 1986, Mathematics, Form and Function, Berlin: Springer-Verlag. (Scholar)
  • Mathias, A.R.D., 2001a, “The Strength of Mac Lane Set Theory”, Annals of Pure and Applied Logic, 110(1–3): 107–234. doi:10.1016/s0168-0072(00)00031-2 (Scholar)
  • –––, 2001b, “Slim Models of Zermelo Set Theory”, The Journal of Symbolic Logic, 66(2): 487–496. doi:10.2307/2695026 (Scholar)
  • McLarty, Colin, 1992, “Failure of Cartesian Closedness in NF”, Journal of Symbolic Logic, 57(2): 555–6. doi:10.2307/2275291 (Scholar)
  • Nelson, Edward, 1977, “Internal Set Theory, a New Approach to Nonstandard Analysis”, Bulletin of the American Mathematical Society, 83(6): 1165–1198. doi:10.1090/s0002-9904-1977-14398-x (Scholar)
  • Quine, W.V.O., 1937, “New Foundations for Mathematical Logic”, American Mathematical Monthly, 44(2): 70–80. doi:10.2307/2300564 (Scholar)
  • –––, 1945, “On Ordered Pairs”, Journal of Symbolic Logic, 10(3): 95–96. doi:10.2307/2267028 (Scholar)
  • Reinhardt, William N., 1970, “Ackermann’s Set Theory Equals ZF”, Annals of Mathematical Logic, 2(2): 189–249. doi:10.1016/0003-4843(70)90011-2 (Scholar)
  • Robinson, Abraham, 1966, Non-standard Analysis, Amsterdam: North-Holland. (Scholar)
  • Rosser, J. Barkley, 1973, Logic for Mathematicians, second edition, New York: Chelsea. (Scholar)
  • Russell, Bertrand, 1903, The Principles of Mathematics, London: George Allen and Unwin. (Scholar)
  • Specker, Ernst P., 1953, “The Axiom of Choice in Quine’s ‘New Foundations for Mathematical Logic’”, Proceedings of the National Academy of Sciences of the United States of America, 39(9): 972–5. [Specker 1953 available online] (Scholar)
  • Spinoza, Benedict de, 1677, Ethics, reprinted and translated in A Spinoza Reader: the “Ethics” and Other Works, Edwin Curley (ed. and trans.), Princeton: Princeton University Press, 1994. (Scholar)
  • Tupailo, Sergei, 2010, “Consistency of Strictly Impredicative NF and a Little More …”, Journal of Symbolic Logic, 75(4): 1326–1338. doi:10.2178/jsl/1286198149 (Scholar)
  • Vopěnka, Petr, 1979, Mathematics in the Alternative Set Theory, Leipzig: Teubner-Verlag. (Scholar)
  • Wang, Hao, 1970, Logic, Computers, and Sets, New York: Chelsea, p. 406. (Scholar)
  • Whitehead, Alfred North and Bertrand Russell, [PM] 1910–1913, Principia Mathematica, 3 volumes, Cambridge: Cambridge University Press. (Scholar)
  • Wiener, Norbert, 1914, “A Simplification of the Logic of Relations”, Proceedings of the Cambridge Philosophical Society, 17: 387–390. (Scholar)
  • Zermelo, Ernst, 1908, “Untersuchen über die Grundlagen der Mengenlehre I”, Mathematische Annalen, 65: 261–281. (Scholar)

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