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Logica Universalis

, Volume 8, Issue 3–4, pp 499–552 | Cite as

The Scope of Gödel’s First Incompleteness Theorem

  • Bernd Buldt
Article

Abstract

Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of Gödel’s famous first incompleteness theorem.

Mathematics Subject Classification

Primary 03 Secondary 03-02 03A05 03F40 

Keywords

Gödel first incompleteness theorem 

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References

  1. 1.
    Adamowicz Z., Bigorajska T.: Existentially closed structures and Gödel’s second incompleteness theorem. J. Symb. Log. 66, 349–356 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Anderson, A.R., Belnap, N.: Entailment: The Logic of Relevance and Necessity, vol. 1. Princeton University Press, Princeton (1975)Google Scholar
  3. 3.
    Anderson, A.R., Belnap, N., Dunn, M.J.: Entailment: The Logic of Relevance and Necessity, vol. 2. Princeton University Press, Princeton (1992)Google Scholar
  4. 4.
    Artemov, S.N., Beklemishev, L.D.: Provability logic. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 13, pp. 189–360. Springer, Dordrecht (2005)Google Scholar
  5. 5.
    Awodey S.: An answer to Hellman’s question: does category theory provide a framework for mathematical structuralism. Philosophia Mathematica 12, 54–64 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Awodey, S., Coquand, Th., Voevodsky, V. et al.: Homotopy Type Theory: Univalent Foundations of Mathematics. Univalent Foundations Program. Institute for Advanced Study, Princeton (2013)Google Scholar
  7. 7.
    Awodey S., Reck E.H.: Completeness and categoricity. Part I: nineteenth-century axiomatics to twentieth-century metalogic. Hist. Philos. Log. 23, 1–30 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Barwise, J. (ed.): Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics, vol. 90. North-Holland, Amsterdam (1977)Google Scholar
  9. 9.
    Bell, J.L.: Incompleteness in a general setting. Bull. Symb. Log. 13, 21–30 (2007) [14, 122 (2008) (corrections)]Google Scholar
  10. 10.
    Bernays P.: A system of axiomatic set theory—part I. J. Symb. Log. 2, 65–77 (1937)Google Scholar
  11. 11.
    Beklemishev, L.D.: Gödel incompleteness theorems and the limits of their applicability. I. Russ. Math. Surv. 65, 857–899 (2010) [First Russian Uspekhi Matematicheskikh Nauk 65, 61–106 (2010)]Google Scholar
  12. 12.
    Boolos, G.: A new proof of Gödel’s incompleteness theorem. In: [14], pp. 383–388 [First Notices of the American Mathematical Association, vol. 36, pp. 388–390, 676 (1989)]Google Scholar
  13. 13.
    Boolos, G.: The Logic of Provability. Cambridge University Press, Cambridge (1993) [Rev. of The Unprovability of Consistency. An Essay in Modal Logic. Cambridge University Press, Cambridge (1979)]Google Scholar
  14. 14.
    Boolos, G.: Logic, Logic, Logic, with notes by John P. Burgess ed. by Richard Jeffrey. Harvard University Press, Cambridge (1998)Google Scholar
  15. 15.
    Boolos, G., Burgess, J.P., Jeffrey, R.C.: Computability and Logic, 4th edn. Cambridge University Press, Cambridge (2002) [1st edn, 1974]Google Scholar
  16. 16.
    Bovykin, A.: Several proofs of PA-unprovability. In: Blass, A., Zhang, Y. (eds.) Logic and its Applications. International Conference of Logic and its Applications in Algebra and Geometry, April 11–13, 2003. Contemporary Mathematics, vol. 380, pp. 29–43. AMS, Providence (2005)Google Scholar
  17. 17.
    Bovykin, A.: Brief introduction to unprovability. In: Copper, B., et al. (eds.) Logic Colloquium ’06. Lecture Notes in Logic, vol. 32, pp. 38–64. Cambridge University Press, Cambridge (2009)Google Scholar
  18. 18.
    Buechner J.: Are the Gödel incompleteness theorems limitative results for the neurosciences? . J. Biol. Phys. 36, 23–44 (2010)CrossRefGoogle Scholar
  19. 19.
    Buldt, B.: Philosophische Implikationen der Gödelschen Sätze? Ein kritischer Bericht. In: Buldt, B., et al. (eds.) Kurt Gödel. Wahrheit und Beweisbarkeit. Bd. 2: Kompendium zum Werk, pp. 395–438. Hölder-Pichler-Tempsky, Vienna (2002)Google Scholar
  20. 20.
    Buldt, B.: On RC 102-43-14. In: Awodey, S., Klein, C. (eds.) Carnap: From Jena to L.A., pp. 225–246. Open Court, Chicago (2004)Google Scholar
  21. 21.
    Buss, S.R.: Bounded Arithmetic. Bibliopolis, Naples (1986)Google Scholar
  22. 22.
    Buss, S.R.: Bounded arithmetic and propositional proof complexity. In: Schwichtenberg, H. (ed.) Logic of Computation, pp. 67–122. Springer, Berlin (1997)Google Scholar
  23. 23.
    Buss, S.R.: First-order proof theory of arithmetic. In: Buss, S. (ed.) Handbook of Proof Theory, pp. 79–147. North-Holland, Amsterdam (1998)Google Scholar
  24. 24.
    Calude, C.S., Jürgensen, H.: Is complexity a source of incompleteness? Adv. Appl. Math. 35, 1–15 (2005)Google Scholar
  25. 25.
    Carnap, R.: Logische Syntax der Sprache. Schriften zur wissenschaftlichen Weltaufffassung, vol. 8. Springer, Vienna (1934) (2nd edn, 1968)Google Scholar
  26. 26.
    Carnap, R.: Autobiography. In: Schilpp, P.A. (ed.) The Philosophy of Rudolf Carnap. Library of Living Philosophers, vol. XI, pp. 1–84. Open Court, Chicago (1963)Google Scholar
  27. 27.
    Chaitin G.: A theory of program size formally identical to information theory. J. ACM 22, 329–340 (1975)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Chaitin, G.: Algorithmic Information Theory. Cambridge University Press, Cambridge (1987)Google Scholar
  29. 29.
    Chvalovský , Chvalovský : On the independence of axioms in BL and MTL. Fuzzy Sets Syst. 197, 123–129 (2012)CrossRefzbMATHGoogle Scholar
  30. 30.
    Collins G.E., Halpern J.D.: On the interpretability of arithmetic in set theory. Notre Dame J. Form. Log. 11, 477–483 (1970)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Corcoran J., Frank W., Maloney M.: String theory. J. Symb. Log. 39, 635–637 (1974)CrossRefMathSciNetGoogle Scholar
  32. 32.
    da Costa, N.C.A., Krause, D., Bueno, O.: Paraconsistent logics and paraconsistency. In: Gabbay, D.M., Thagard, P., Woods, J. (eds.) Handbook of the Philosophy of Science [Jacquette, D. (ed.) Philosophy of Logic, vol. 5], pp. 655–781. Elsevier, Amsterdam (2006)Google Scholar
  33. 33.
    Davis, M. (ed.): The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems, and Computable Functions. Raven P, Hewlett (1965)Google Scholar
  34. 34.
    Davis, M., Matiyasevich, Y., Robinson, J.: Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solution. In: Browder, F.E. (ed.) Mathematical Developments Arising From Hilbert’s Problems. Proceedings of Symposia in Pure Mathematics, vol. 28, pp. 223–378 AMS, Providence (1976)Google Scholar
  35. 35.
    Davis M., Putnam H., Robinson J.: The decision problem for exponential Diophantine equations. Ann. Math. 74, 425–436 (1961)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Dawes, A.M., Florence, J.B.: Independent Gödel sentences and independent sets. J. Symb. Log. 40, 159–166 (1975)Google Scholar
  37. 37.
    Dedekind, R.: Was sind und was sollen die Zahlen. In: [39], Bd. 3, pp. 335–390 [First Vieweg, Braunschweig (1888); engl. in [38], pp. 31–115]Google Scholar
  38. 38.
    Dedekind, R.: Essays on the Theory of Number. Open Court, Chicago (1901) [Transl. by Wooster Woodruff Beman]Google Scholar
  39. 39.
    Dedekind, R.: In: Fricke, R., Noether, E., Ore, Ö. (eds.) Gesammelte mathematische Werke, vols. 1–3. Vieweg, Braunschweig (1930–1932)Google Scholar
  40. 40.
    Ehrenfeucht, A.: Separable theories. In: Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 14, pp. 17–19 (1961)Google Scholar
  41. 41.
    Feferman S.: Transfinite recursive progressions of axiomatic theories. J. Symb. Log. 27, 259–316 (1962)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Feferman, S.: Turing in the land of O(z). In: Herkin, R. (ed.) The Universal Turing Machine. A Half-Century Survey, pp. 113–147. Kammerer & Unverzagt, Hamburg (1988)Google Scholar
  43. 43.
    Feferman S.: Reflecting on incompleteness. J. Symb. Log. 56, 1–49 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  44. 44.
    Feferman, S., Solovay, R.M.: Introductory Note to 1972a. Remark 2 in [69], pp. 287–292Google Scholar
  45. 45.
    Feferman S., Spector C.: Incompleteness along paths in progressions of theories. J. Symb. Log. 27, 383–390 (1962)CrossRefMathSciNetGoogle Scholar
  46. 46.
    Franzén T.: Transfinite progressions: a second look at completeness. Bull. Symb. Log. 10, 367–389 (2004)CrossRefzbMATHGoogle Scholar
  47. 47.
    Franzén, T.: Gödel’s Theorem: An Incomplete Guide to its Use and Abuse. A K Peters, Wellesley (2005)Google Scholar
  48. 48.
    Friedman, H.: Concrete incompleteness from EFA through large cardinals (slides, 05/10/2010). http://www.math.ohio-state.edu/~friedman/
  49. 49.
    Friedman, H.: My forty years on his shoulders. In: Baaz, M., et al. (eds.) Kurt Gödel and the Foundations of Mathematics: Horizons of Truth, pp. 399–432. Cambridge University Press, Cambridge (2011)Google Scholar
  50. 50.
    Friedman, H., Meyer, R.K.: Whither relevant arithmetic? J. Symb. Log. 57, 824–831 (1992)Google Scholar
  51. 51.
    Friedman, H., Visser, A.: When Bi-interpretability Implies Synonymy, Logic Group Preprint Series, vol. 320. Utrecht (2014)Google Scholar
  52. 52.
    Ganea M.: Arithmetic on semigroups. J. Symb. Log. 74, 265–278 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  53. 53.
    Givant, S., Tarski, A.: Peano arithmetic and the Zermelo-like theory of sets with finite ranks (abstract 77T-E51). Notices of the American Mathematical Society, vol. 24, A-437 (1977)Google Scholar
  54. 54.
    Gödel, K.: Über die Vollständigkeit des Logikkalküls (unpublished dissertation), German-English in [68], pp. 60–101Google Scholar
  55. 55.
    Gödel, K.: Über die Vollständigkeit der Axiome des logischen Funktionenkalküls, German-English in [68], pp. 102–132 [First Monatshefte für Mathe-matik und Physik 37, 349–360 (1930)]Google Scholar
  56. 56.
    Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, German-English in [67], pp. 144–195 [First Monatshefte für Mathematik und Physik 38, 173–198 (1931)]Google Scholar
  57. 57.
    Gödel, K.: Über Vollständigkeit und Widerspruchsfreiheit, German-English in [67], pp. 234–237 [First Ergebnisse eines mathematischen Kolloquiums 3, 12–13 (1932)]Google Scholar
  58. 58.
    Gödel, K.: Eine Interpretation des intuitionistischen Aussagenkalküls, German-English in [68], pp. 300–303 [First Ergebnisse eines mathematischen Kolloquiums 4, 39–40 (1933)]Google Scholar
  59. 59.
    Gödel, K.: On Undecidable Propositions of Formal Mathematical Systems (Lecture Notes, 1934), in [68], pp. 346–371 [First in [33], 39–74]Google Scholar
  60. 60.
    Gödel, K.: Über die Länge von Beweisen, German-English in [68], pp. 396–399 [First Ergebnisse eines mathematischen Kolloquiums 7, 23–24 (1936)]Google Scholar
  61. 61.
    Gödel, K.: What is Cantor’s continuum problem in [69], pp. 176–187 [First American Mathematical Monthly 54, 515–525, (1947), 55, 151 (1948)]Google Scholar
  62. 62.
    Gödel, K.: Some basic theorems on the foundations of mathematics and their implications (Gibbs Lecture), posthumously in [70], pp. 304–323Google Scholar
  63. 63.
    Gödel, K.: Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, in [69], pp. 240–251 [First Dialectica 12, 280–287 (1958)]Google Scholar
  64. 64.
    Gödel, K.: What is Cantor’s continuum problem, in [69], pp. 254–270 [Rev. version of [61Google Scholar
  65. 65.
    Gödel, K.: Postscriptum (3 June 1964), in [68], pp. 369–371 [Further editorial corrections to [59] can be found throughout the reprint]Google Scholar
  66. 66.
    Gödel, K.: Remark by the author (18 May 1966), in [68], p. 235 [First in [191], footnote to the reprint of [57Google Scholar
  67. 67.
    Gödel, K.: Appendix, posthumously in [69], pp. 305–306 [An appendix to the galley proofs of the English translation of [63Google Scholar
  68. 68.
    Gödel, K.: Collected Works. In: Feferman, S., et al. (eds.) Publications 1929–1936, vol. 1. Oxford University Press, Oxford (1986)Google Scholar
  69. 69.
    Gödel, K.: Collected Works. In: Feferman, S., et al. (eds.) Publications 1938–1974, vol. 2. Oxford University Press, Oxford (1990)Google Scholar
  70. 70.
    Gödel, K.: Collected Works. In: Feferman, S., et al. (eds.) Unpublished Essays and Lectures, vol. 3. Oxford University Press, Oxford (1995)Google Scholar
  71. 71.
    Gödel, K.: Collected Works. In: Feferman, S., et al. (eds.) Correspondence A-G, vol. 4. Oxford University Press, Oxford (2003)Google Scholar
  72. 72.
    Goldfarb, W.: On the effective ω-rule, Zeitschrift für mathematische Logik und Grundlagen der Mathematik (Mathematical Logic Quarterly), vol. 21, pp. 409–412 (1975)Google Scholar
  73. 73.
    Goldfarb, W.: Herbrand’s theorem and the incompleteness of arithmetic. Iyyum. A Jerusalem Philosophical Quarterly, vol. 39, pp. 45–64 (1990)Google Scholar
  74. 74.
    Grzegorczyk A.: Undecidability without arithmetization. Stud. Log. 79, 163–230 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  75. 75.
    Grzegorczyk A., Mostowski A., Ryll-Nardzewski C.: The classical and ω-complete arithmetic. J. Symb. Log. 23, 188–206 (1958)CrossRefMathSciNetGoogle Scholar
  76. 76.
    Grzegorczyk, A., Zdanowski, K.: Undecidability and concatenation. In: Ehrenfeucht, A., Marek, V.W., Srebrny M. (eds.) Andrzej Mostowski and Foundational Studies, pp. 72–91. IOS P, Amsterdam, (2008)Google Scholar
  77. 77.
    Halbach, V., Visser, A.: Self-reference in arithmetic (manuscript, 12/15/2013)Google Scholar
  78. 78.
    Hájek, P.: Metamathematics of Fuzzy Logics. In: Trends in Logic-Studia Logica Library, vol. 4. Kluwer, Dordrecht (1998)Google Scholar
  79. 79.
    Hájek P.: Mathematical fuzzy logic and natural numbers. Fundamenta Informaticae 81, 155–163 (2007)MathSciNetzbMATHGoogle Scholar
  80. 80.
    Hájek P.: Towards metamathematics of weak arithmetics over fuzzy logic. Log. J. IGPL 19, 467–475 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  81. 81.
    Hájek, P., Pudlák, P.: Metamathematics of First-Order Arithmetic. Perspectives in Mathematical Logic. Springer, Berlin (1993)Google Scholar
  82. 82.
    Harrington, L.A., Morley, M.D., Scedrov, A., Simpson, S.G. (eds): Harvey Friedman’s Research on the Foundations of Mathematics. Studies in Logic and the Foundations of Mathematics, vol. 117. North-Holland, Amsterdam (1985)Google Scholar
  83. 83.
    Henkin L.: A generalization of the concept of ω-completeness. J. Symb. Log. 19, 183–196 (1954)CrossRefMathSciNetzbMATHGoogle Scholar
  84. 84.
    Herbrand, J.: On the consistency of arithmetic, in [85], pp. 282–297 [English transl. Sur la non-contradiction de l’arithmétique, Journal für reine and angewandte Mathematik 166, 1–8 (1931)]Google Scholar
  85. 85.
    Herbrand, J.: Logical Writings. In: Goldfarb, W.D. (ed.) Harvard University Press, Cambridge (1971)Google Scholar
  86. 86.
    Hilbert, D., Bernays, P.: Grundlagen der Mathematik, Bd. 2. Springer, Berlin (1939)Google Scholar
  87. 87.
    Hintikka, J.: Principles of Mathematics Revisited. Cambridge University Press, Cambridge (1996)Google Scholar
  88. 88.
    Hintikka, J., Sandu, G.: A revolution in logic? Nord. J. Philos. Log. 1, 169–183 (1996)Google Scholar
  89. 89.
    Isaacson, D.: Necessary and sufficient conditions for undecidability of the Gödel sentence and its truth. In: DeVidi, D., Hallet, M., Clark, P. (eds.) Logic, Mathematics, Philosophy: Vintage Enthusiasm. Essays in Honour of John L. Bell. The Western Ontario Series in Philosophy of Science, vol. 75, pp. 135–152. Springer, Dordrecht (2011)Google Scholar
  90. 90.
    Japaridze (Dzhaparidze), G.: Introduction to computability logic. Ann. Pure Appl. Log. 123, 1–99 (2003)Google Scholar
  91. 91.
    Japaridze (Dzhaparidze), G.: PTArithmetic, The Baltic International Yearbook of Cognition, Logic and Communication. Games, Game Theory and Game Semantics, vol. 8, pp. 1–186 (2013)Google Scholar
  92. 92.
    Jones J.P.: Three universal representations of recursively enumerable sets. J. Symb. Log. 43, 335–351 (1978)CrossRefzbMATHGoogle Scholar
  93. 93.
    Jones J.P.: Universal Diophantine equation. J. Symb. Log. 47, 549–571 (1982)CrossRefzbMATHGoogle Scholar
  94. 94.
    Jones, J.P., Shepherdson, J.C.: Variants of Robinson’s essentially undecidable theory R. Arch. Math. Log. 23, 61–64 (1983)Google Scholar
  95. 95.
    Kikuchi, M.: A note on Boolos’ proof of the incompleteness theorem. Math. Log. Q. 40, 528–532 (1994)Google Scholar
  96. 96.
    Kirby, L., Paris, J.: Accessible independence results for Peano arithmetic. Bull. Lond. Math. Soc. 14, 285–293 (1982)Google Scholar
  97. 97.
    Kleene, S.C.: A symmetric form of Gödel’s theorem. Indagationes Mathematicae 12, 244–246 [Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings of the Section of Sciences 53 (1950), 800–802 (1950)]Google Scholar
  98. 98.
    Kleene, S.C.: Introduction to Metamathematics. North-Holland, Amsterdam (1952)Google Scholar
  99. 99.
    Kleene, S.C.: Introductory note to 1934. In: [68], pp. 338–345Google Scholar
  100. 100.
    Kochen, S., Kripke, S.: Non-standard models of Peano arithmetic, L’Enseignement Mathématique, (2) 28, 211–231 (1982) [First Logic and Algorithmic. An International Symposium in Honour of E. Specker, Monographie de L’Enseignement Mathèmatique, vol. 30, pp. 275–295. Gèneve University Press, Geneva (1982)]Google Scholar
  101. 101.
    Kotlarski, H.: On the incompleteness theorems. J. Symb. Log. 59, 1414–1419 (1994)Google Scholar
  102. 102.
    Kotlarski, H.: An addition to Rosser’s theorem. J. Symb. Log. 61, 285–292 (1996)Google Scholar
  103. 103.
    Kotlarski, H.: Other proofs of old results. Math. Log. Q. 44, 474–480 (1998)Google Scholar
  104. 104.
    Kotlarski, H.: The incompleteness theorems after 70 years. Ann. Pure Appl. Log. 126, 125–138 (2004)Google Scholar
  105. 105.
    Kreisel, G.: A refinement of ω-consistency (abstract). J. Symb. Log. 22, 108–109 (1957)Google Scholar
  106. 106.
    Kreisel, G.: Hilbert’s programme. Dialectica 12, 346–372 (1958) [Rev. In: Benecerraf, P., Putnam, H. (eds.) Philosophy of Mathematics: Selected Readings, pp. 157–180. Prentice Hall/Blackwell, Englewood Cliffs/Oxford (1964)]Google Scholar
  107. 107.
    Kreisel, G.: Some reasons for generalizing recursion theory. In: Gandy, R.O., Yates, C.E.M. (eds.) Logic Colloquium ’69, Proceedings of the Summer School and Colloquium in Mathematical Logic, Manchester, August 1969. Studies in Logic and the Foundations of Mathematics, vol. 61, pp. 139–198. North-Holland, Amsterdam (1971)Google Scholar
  108. 108.
    Kreisel, G.: Kurt Gödel, 28 April 1906–14 January 1978. Biographical Memoirs of Fellows of the Royal Society of London 26, 149–224 (1980) [Corrections, ibid. 27, 697; 28, 718]Google Scholar
  109. 109.
    Kripke, S.A.: “Flexible” predicates of formal number theory. Proc. Am. Math. Soc. 13, 647–650 (1962)Google Scholar
  110. 110.
    Kröger F.: On the interpretability of arithmetic in temporal logic. Theor. Comput. Sci. 73, 47–60 (1990)CrossRefzbMATHGoogle Scholar
  111. 111.
    Lambek J.: How to program an infinite abacus. Can. Math. Bull. 4, 295–302 (1961)CrossRefMathSciNetzbMATHGoogle Scholar
  112. 112.
    Lambek J.: What is the world of mathematics. Ann. Pure Appl. Log. 126, 149–158 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  113. 113.
    Lambek, J., Scott, P.J.: Reflections on the categorical foundations of mathematics. In: Sommaruga, G. (ed.) Foundational Theories of Classical and Constructive Mathematics. Reflections on a Categorical Foundations of Mathematics, Western Ontario Series in Philosophy of Science, vol. 76, pp. 171–186. Springer, Berlin (2011)Google Scholar
  114. 114.
    Landry, E., Marquis, J.-P.: Categories in context: historical, foundational, and philosophical. Philosophia Mathematica (III) 13, 1–43 (2005)Google Scholar
  115. 115.
    Lawvere, F.W.: An elementary theory of the category of sets. Theory Appl. Categ. 11, 1–35 (2005) [First Proceedings of the National Academy of Science of the USA, vol. 52, pp. 1506–1511 (1964)]Google Scholar
  116. 116.
    Lawvere, F.W.: Diagonal arguments and Cartesian closed categories. Theory Appl. Categ. 15, 1–13 (2006) Category Theory, Homology Theory and Their Applications. Proceedings of the Conference Held at the Seattle Research Center of the Battelle Memorial Institute, June 24–July 19, 1968. [In: Hilton, P.J. (ed.)] Lecture Notes in Mathematics, vol. 92, pp. 134–145. Springer, Berlin (1969)]Google Scholar
  117. 117.
    Mac Lane, S.: Mathematics, Form and Function. Springer, New York (1986)Google Scholar
  118. 118.
    Macintyre, A.: The impact of Gödel’s incompleteness theorems on mathematics. In: Baaz, M. et al. (eds.) Kurt Gödel and the Foundations of Mathematics: Horizons of Truth, pp. 3–25. Cambridge University Press, Cambridge (2011)Google Scholar
  119. 119.
    Maietti M.E.: Joyal’s arithmetic universe as list-arithmetic pretopos. Theory Appl. Categ. 24, 39–83 (2010)MathSciNetzbMATHGoogle Scholar
  120. 120.
    Marquis J.-P.: Categorical foundations of mathematics. Or how to provide foundations for abstract mathematics. Rev. Symb. Log. 6, 51–75 (2013)MathSciNetzbMATHGoogle Scholar
  121. 121.
    Matiyasevich, Y.V.: Enumerable sets are Diophantine. Dokl. Math. 11, 354–357 (1970) [First Russian Doklady Akademii Nauk SSSR (Proceedings of the Russian Academy of Sciences) 191, 279–282 (1970)]Google Scholar
  122. 122.
    Matiyasevich, Y.V.: Hilbert’s Tenth Problem. Foundations of Computing. MIT Press, Cambridge (1993)Google Scholar
  123. 123.
    Meyer, R.K.: ⊃-E is admissible in “true” relevant arithmetic. J. Philos. Log. 27, 327–351 (1998)Google Scholar
  124. 124.
    Meyer, R.K., Mortensen, C.: Inconsistent models for relevant arithmetics. J. Symb. Log. 49, 917–929 (1984)Google Scholar
  125. 125.
    Monk, J.D.: Mathematical Logic. Graduate Texts in Mathematics. Springer, New York (1976)Google Scholar
  126. 126.
    Montague, R.: The continuum of relative interpretability (abstract). J. Symb. Log. 23, 460 (1958)Google Scholar
  127. 127.
    Montague, R.: Theories incomparable with respect to relative interpretability. J. Symb. Log. 27, 195–211 (1962)Google Scholar
  128. 128.
    Montagna F., Mancini A.: A minimal predicative set theory. Notre Dame J. Form. Log. 35, 186–203 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  129. 129.
    Mortensen C.: Inconsistent number systems. Notre Dame J. Form. Log. 29, 45–60 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  130. 130.
    Mortensen, C.: Inconsistent Mathematics. Mathematics and Its Applications, vol. 312. Kluwer, Dordrecht (1995)Google Scholar
  131. 131.
    Mortensen, C.: Inconsistent mathematics. Some philosophical implications. In: Gabbay, D.M., Thagard, P., Woods, J. (eds.) Handbook of the Philosophy of Science [Irvine, A.I. (ed.) Philosophy of Mathematics, vol. 4], pp. 631–649. Elsevier, Amsterdam (2009)Google Scholar
  132. 132.
    Mostowski, A.: A generalization of the incompleteness theorem. In: [133], vol. 2, pp. 376–403 [First Fundamenta Mathematicae 49, 205–232 (1961)]Google Scholar
  133. 133.
    Mostowksi, A.: Foundational Studies. In: Selected Works. Kuratowski, K., et al. (eds.) Studies in Logic and the Foundations of Mathematics, vol. 93 North-Holland & Warsaw/Pánstowe Wydawnictwo Naukowe, Amsterdam (1979)Google Scholar
  134. 134.
    Mostowski, A., Robinson, R.M., Tarski, A.: Undecidability and essential undecidability in arithmetic. In: [187], pp. 37–87Google Scholar
  135. 135.
    Murawksi, R.: Undefinability of truth. The problem of the priority: Tarski vs. Gödel. Hist. Philos. Log. 19, 153–160 (1998)Google Scholar
  136. 136.
    Myhill, J.: An absolutely independent set of \({\Sigma^0_1}\) -sentences. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 18, 107–109 (1972)Google Scholar
  137. 137.
    Nelson, E.: Predicative Arithmetic. Mathematical Notes, vol. 32. Princeton University Press, Princeton (1986)Google Scholar
  138. 138.
    Odifreddi, P.: Classical Recursion Theory. [I:] The Theory of Functions and Sets of Natural Numbers. Studies in Logic and the Foundations of Mathematics, vol. 125. North-Holland, Amsterdam (1989)Google Scholar
  139. 139.
    Orey S.: On ω-consistency and related properties. J. Symb. Log. 21, 246–252 (1956)CrossRefMathSciNetzbMATHGoogle Scholar
  140. 140.
    Owings, J.C., Jr: Diagonalization and the recursion theorem. Notre Dame J. Form. Log. 14, 95–99 (1973)Google Scholar
  141. 141.
    Paris, J.B., Harrington, L.: A mathematical incompleteness in Peano Arithmetic. In: [8], pp. 1133–1142Google Scholar
  142. 142.
    Paris J.B., Pathmanathan N.: A note on Priest’s finite inconsistent arithmetics. J. Philos. Log. 35, 529–537 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  143. 143.
    Paris J.B., Sirokofskich A.: On LP-models of arithmetic. J. Symb. Log. 73, 212–226 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  144. 144.
    Paulson, L.C.: A machine-assisted proof of Gödel’s incompleteness theorems for the theory of hereditarily finite sets. Rev. Symb. Log. 7, 484–498 (2014)Google Scholar
  145. 145.
    Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, Comptes-rendus du I Congrès des Mathématiciens des Pays Slaves (Warszawa 1929). Warsaw, pp. 92–101, 395 (1930)Google Scholar
  146. 146.
    Priest G.: The logic of paradox. J. Philos. Log. 8, 219–241 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  147. 147.
    Priest G.: Minimally inconsistent LP. Stud. Log. 50, 321–331 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  148. 148.
    Priest, G.: Is arithmetic consistent? Mind NS 103, 337–349 (1994)Google Scholar
  149. 149.
    Priest G.: Inconsistent models of arithmetic. Part I: finite models. J. Philos. Log. 26, 223–235 (1997)MathSciNetzbMATHGoogle Scholar
  150. 150.
    Priest G.: Inconsistent models of arithmetic. Part II: the general case. J. Symb. Log. 65, 1519–1529 (2000)MathSciNetzbMATHGoogle Scholar
  151. 151.
    Priest, G.: In Contradiction: A Study of the Transconsistent. Oxford University Press, Oxford (2006) [1st edn. Nijhoff, Dordrecht (1987)]Google Scholar
  152. 152.
    Putnam, H.: Non-standard numbers and Kripke’s proof of the Gödel theorem. Notre Dame J. Form. Log. 41, 53–58 (2000)Google Scholar
  153. 153.
    Quinsey, J.E.: Some Problems in Logic. Doctoral dissertation. St. Catherine’s College, Oxford (1980)Google Scholar
  154. 154.
    Raatikainen, P.: On interpreting Chaitin’s incompleteness theorem. J. Philos. Log. 27, 569–586 (1998)Google Scholar
  155. 155.
    Raatikainen, P.: On the philosophical relevance of Gödel’s incompleteness theorems. Revue Internationale de Philosophie 59, 513–534 (2005)Google Scholar
  156. 156.
    Reid, S.: Relevant Logic. Blackwells, Oxford (1988)Google Scholar
  157. 157.
    Reidhaar-Olson, L.: A new proof of the fixed-point theorem of provability logic. Notre Dame J. Form. Log. 31, 37–43 (1990)Google Scholar
  158. 158.
    Restall, G.: Models for substructural arithmetics. In: Bilková, M. (ed.) Miscellanea Logica, pp. 1–20. Charles University, Prague (2008)Google Scholar
  159. 159.
    Robinson J.: Definability and decision problems in arithmetic. J. Symb. Log. 14, 98–114 (1949)CrossRefzbMATHGoogle Scholar
  160. 160.
    Robinson, R.M.: An essentially undecidable axiom system. In: Proceedings of the International Congress of Mathematicians, Cambridge, MA, August 30–September 6, 1950, vol. 1, pp. 729–730. AMS, Providence (1952)Google Scholar
  161. 161.
    Rose, H.E.: Subrecursion: Functions and Hierarchies. Oxford Logic Guides, vol. 9. Oxford University Press, Oxford (1984)Google Scholar
  162. 162.
    Rosser, B.: Extensions of some theorems of Gödel and Church. J. Symb. Log. 1, 87–91 (1936)Google Scholar
  163. 163.
    Rosser B.: Gödel theorems for non-constructive logics. J. Symb. Log. 2, 129–137 (1937)CrossRefGoogle Scholar
  164. 164.
    Sacks, G.E.: Higher Recursion Theory. Perspectives in Mathematical Logic. Springer, Berlin (1990)Google Scholar
  165. 165.
    Schiller, F.: Xenien. In: Petersen, J. (ed.) Schillers Werke, Blumenthal, L. (ed.) Gedichte in der Reihenfolge ihres Erscheinens, vol. 1, 1776–1799, Böhlau, Weimar (1943)Google Scholar
  166. 166.
    Schmerl, U.R.: Iterated reflection principles and the ω-rule. J. Symb. Log. 47, 721–733 (1982)Google Scholar
  167. 167.
    Scholz, H.: Mathesis Universalis. Abhandlungen zur Philosophie als strenger Wissenschaft. In: Kambartel, H., Kambartel, F., Ritter, J. (eds.) Schwabe, Basel (1961)Google Scholar
  168. 168.
    Seising, R.: Fuzzification of Systems: The Genesis of Fuzzy Set Theory and its Initial Applications—Developments up to the 1970s. Studies in Fuzziness and Soft Computing, vol. 21. Springer, Berlin (2007) [First German edn. Steiner, Stuttgart (2005)]Google Scholar
  169. 169.
    Shoenfield, J.R.: On a restricted ω-rule. Bulletin de l’Académie Polonaise des Sciences 7, 405–407 (1959)Google Scholar
  170. 170.
    Skolem, T.: Über einige Satzfunktionen in der Arithmetik. In: [171], pp. 281–306 [First Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo, I. Matematisk-naturvidenskabelig klasse, vol. 7, pp. 1–28 (1931)]Google Scholar
  171. 171.
    Skolem, T.: Selected works in logic. In: Fenstad, J.E. (ed.) Universitetsforlaget, Oslo (1970)Google Scholar
  172. 172.
    Smith, P.: An Introduction to Gödel’s Theorems. Cambridge University Press, Cambridge (2007)Google Scholar
  173. 173.
    Smoryński, C.: Avoiding self-referential statements. Proc. Am. Math. Soc. 70, 181–184 (1978)Google Scholar
  174. 174.
    Smoryński, C.: Some rapidly growing functions. Mathematical Intelligencer 2, 149–154 (1979–1980) [repr. in [82], 367–380]Google Scholar
  175. 175.
    Smoryński, C.: Fifty years of self-reference in arithmetic. Notre Dame J. Form. Log. 22, 357–374 (1981)Google Scholar
  176. 176.
    Smoryński, C.: The varieties of arboreal experience. Mathematical Intelligencer 4, 182–189 (1982) [repr. in [82], pp. 381–398]Google Scholar
  177. 177.
    Smoryński, C.: Self-reference and Modal Logic. Universitext. Springer, New York (1985)Google Scholar
  178. 178.
    Smullyan R.M.: Languages in which self-reference is possible. J. Symb. Log. 22, 55–67 (1957)CrossRefMathSciNetzbMATHGoogle Scholar
  179. 179.
    Smullyan R.M.: Chameleonic languages. Synthese 60, 201–224 (1984)CrossRefMathSciNetGoogle Scholar
  180. 180.
    Smullyan, R.M.: Diagonalization and Self-Reference. Oxford Logic Guides, vol. 27. Oxford University Press, New York (1994)Google Scholar
  181. 181.
    Solovay, R.M.: Injecting Inconsistencies into models of PA. Ann. Pure Appl. Log. 44, 101–132 (1989)Google Scholar
  182. 182.
    Švejdar, V.: An interpretation of Robinson arithmetic in its Grzegorczyk’s weaker variant. Fundamenta Informaticae 81, 347–354 (2007)Google Scholar
  183. 183.
    Švejdar, V.: Weak theories and essential incompleteness. In: Peliš, M. (ed.) The Logica Yearbook 2007: Proceedings of the Logica 07 International Conference, pp. 213–224. Philosophia, Prague (2008)Google Scholar
  184. 184.
    Švejdar, V.: On interpretability in the theory of concatenation. Notre Dame J. Form. Log. 50, 87–95 (2009)Google Scholar
  185. 185.
    Świerczkowski, S.: Finite sets and Gödel’s incompleteness theorems, Dissertationes Mathematicae Rozprawy Matematyczne, vol. 422. Institute of Mathematics, Polish Academy of Sciences, Warsaw (2003)Google Scholar
  186. 186.
    Szmielew, W., Tarski, A.: Mutual interpretability of some essentially undecidable theories. In: Proceedings of the International Congress of Mathematicians, Cambridge, MA, August 30–September 6, 1950, vol. 1, p. 734. AMS, Providence (1952)Google Scholar
  187. 187.
    Tarski, A., Mostowski, A., Robinson, R.M.: Undecidable Theories. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1953) [3rd edn, 1971]Google Scholar
  188. 188.
    Tarski, A., Givant, S.: A formalization of set theory without variables. American Mathematical Society Colloquium Publications, vol. 41. AMS, Providence (1987)Google Scholar
  189. 189.
    Turing, A.: Systems of logic based on ordinals. In: [68], pp. 155–222 [First Proceedings of the London Mathematical Society (2) 45, 161–228 (1939)]Google Scholar
  190. 190.
    van Bendegem, J.P.: In defense of strict finitism. Math. Constr. 7, 141–149 (2013)Google Scholar
  191. 191.
    van Heijenoort, J. (ed.): From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge (1967)Google Scholar
  192. 192.
    van Lambalgen, M.: Algorithmic information theory. J. Symb. Log. 54, 1389–1400 (1989)Google Scholar
  193. 193.
    Vaught, R.L.: On a theorem of Cobham concerning undecidable theories. In: Nagel, E., Suppes, P., Tarski, A. (eds.) Logic, Methodology, and Philosophy of Science. Proceedings of the 1960 International Congress, pp. 14–25. Stanford University Press, Stanford (1962)Google Scholar
  194. 194.
    Visser, A.: Growing commas: a study of sequentiality and concatenation. Notre Dame J. Form. Log. 50, 61–85 (2009)Google Scholar
  195. 195.
    Visser, A.: Why the theory R is special. In: Tennant, N. (ed.) Foundational Adventures. Essays in honour of Harvey Friedman, Tributes, vol. 22. College Publications, London (2014) [Copy used: Logic Group Preprint Series, vol. 279, Utrecht (2009)]Google Scholar
  196. 196.
    Wang H.: Undecidable sentences generated by semantic paradoxes. J. Symb. Log. 20, 31–43 (1952)CrossRefGoogle Scholar
  197. 197.
    Wilkie A.J., Paris J.B.: On the scheme of induction for bounded formulas. Ann. Pure Appl. Log. 35, 261–302 (1987)CrossRefMathSciNetzbMATHGoogle Scholar
  198. 198.
    Yanofsky, N.Y.: A universal approach to self-referential paradoxes, incompleteness and fixed points. Bull. Symb. Log. 9, 362–386 (2003)Google Scholar
  199. 199.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)Google Scholar
  200. 200.
    Zadeh L.A.: Fuzzy logic, neural networks, and soft computing. Commun. ACM 37, 77–84 (1994)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of PhilosophyIndiana University-Purdue University Fort Wayne (IPFW)Fort WayneUSA

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