Skip to main content Accessibility help
×
Home

INCOMPLETENESS VIA PARADOX AND COMPLETENESS

  • WALTER DEAN (a1)

Abstract

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth.

Copyright

Corresponding author

*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF WARWICK COVENTRY CV4 7AL, UK E-mail: W.H.Dean@warwick.ac.uk

References

Hide All
Page references to Hilbert & Bernays, Grundlagen der Mathematik, Volumes I and II are in the form p. m/n for m the page in the first edition and n the page in the second edition.
Ackermann, W. (1928). Über die Erfüllbarkeit gewisser Zählausdrücke. Mathematische Annalen, 100 (1), 638649.
Bernays, P. (1930). Die Philosophie der Mathematik und die Hilbertsche Beweistheorie. Blätter für deutsche Philosophie, 4, 326367. Reprinted in Bernays (1976), pp. 17–62 and in Mancosu (1998), pp. 234–265.
Bernays, P. (1937). A system of axiomatic set theory: Part I. Journal of Symbolic Logic, 2(1), 6577.
Bernays, P. (1942). A system of axiomatic set theory: Part III. Infinity and enumerability. Analysis. Journal of Symbolic Logic, 7(2), 6589.
Bernays, P. (1954a). A system of axiomatic set theory: Part VII. Journal of Symbolic Logic, 19(2), 8196.
Bernays, P. (1954b). Zur Beurteilung der situation in der beweistheoretischen Forschung. Revue Internationale de Philosophie, 27/28, 913.
Bernays, P. (1970). Die schematische Korrespondenz und die idealisierten Strukturen. Dialectica, 24, 5366. Reprinted in Bernays (1976), pp. 176–188.
Bernays, P. (1976). Abhandlungen zur Philosophie der Mathematik. Darmstadt: Wiss. Buchgesellschaft.
Bernays, P. & Fraenkel, A. (1958). Axiomatic Set Theory. Amsterdam: North-Holland.
Boolos, G. (1989). A new proof of the Gödel incompleteness theorem. Notices of the American Mathematical Society, 36(4), 388390.
Borel, É. (1898). Leçons sur la théorie des fonctions. Paris: Gauthier-Villars et fils.
Church, A. (1956). Introduction to Mathematical Logic. Princeton: Princeton University Press.
Church, A. (1976). Comparison of Russell’s resolution of the semantical antinomies with that of Tarski. The Journal of Symbolic Logic, 41(4), 747760.
Cieśliński, C. (2002). Heterologicality and incompleteness. Mathematical Logic Quarterly, 48(1), 105110.
Cieśliński, C. (2018). The Epistemic Lightness of Truth: Deflationism and its Logic. Cambridge: Cambridge University Press.
Cohen, P. (1966). Set Theory and the Continuum Hypothesis. New York: W.A. Benjamin.
Dean, W. (2015). Arithmetical reflection and the provability of soundness. Philosophia Mathematica, 23(1), 3164.
Dean, W. (2017). Bernays and the completeness theorem. Annals of the Japanese Association for the Philosophy of Science, 25, 4455.
Dean, W. & Walsh, S. (2017). The prehistory of the subsystems of second-order arithmetic. The Review of Symbolic Logic, 10(2), 357396.
Doets, K. (1999). Relatives of the Russell paradox. Mathematical Logic Quarterly, 45, 7383.
Dummett, M. (1978). Frege’s distinction between sense and reference. Truth and Other Enigmas. Cambridge, MA: Harvard University Press, pp. 116144.
Ebbs, G. (2015). Satisfying predicates: Kleene’s proof of the Hilbert–Bernays theorem. History and Philosophy of Logic, 36(4), 346366.
Enayat, A. & Visser, A. (2015). New constructions of satisfaction classes. In Achourioti, T., Galinon, H., Fernández, J. M., and Fujimoto, K., editors. Unifying the Philosophy of Truth. Dordrecht: Springer, pp. 321335.
Enayat, A., Łełyk, M., & Wcisło, B. (2019). Truth and feasible reducibility. The Journal of Symbolic Logic, to appear.
Ewald, W. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press.
Ewald, W. & Sieg, W. (editors) (2013). David Hilbert’s Lectures on the Foundations of Logic and Arithmetic 1917–1933. Berlin: Springer.
Feferman, S. (1960). Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49, 3592.
Feferman, S. (1991). Reflecting on incompleteness. Journal of Symbolic Logic, 56(1), 149.
Feferman, S., Dawson, J. W. Jr., Kleene, S. C., Moore, G. H., Solovay, R. M., & van Heijenoort, J. (editors) (1986). Kurt Gödel Collected Works. Vol. I. Publications 1929–1936. Oxford: Oxford Univeristy Press.
Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., & Sieg, W. (editors) (2003). Kurt Gödel Collected Works. Vol. IV. Publications Correspondence A-G. Oxford: Oxford Univeristy Press.
Field, H. (2008). Saving Truth from Paradox. Oxford: Oxford University Press.
Fischer, M., Horsten, L., & Nicolai, C. (2019). Hypatia’s silence: Truth, justification, and entitlement. Noûs, to appear.
Fitch, F. (1952). Symbolic Logic. New York: The Ronald Press Company.
Gentzen, G. (1936). Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, 112, 493565.
Gödel, K. (1930). The completeness of the axioms of the functional calculus of logic. pp. 103123. Reprinted in Feferman et al. (1986).
Gödel, K. (1931a). Correspondence with Ernest Zermelo. Reprinted in Feferman et al. . (2003).
Gödel, K. (1931b). On formally undecidable propositions of Principia Mathematica and related systems I. Reprinted in Feferman et al. (1986).
Gödel, K. (1940). The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory. Princeton: Princeton University Press.
Hájek, P. & Pudlák, P. (1998). Metamathematics of First-Order Arithmetic (first edition 1993). Berlin: Springer.
Halbach, V. (2011). Axiomatic Theories of Truth. Cambridge: Cambridge University Press.
Halbach, V. & Horsten, L. (2006). Axiomatizing Kripke’s theory of truth. The Journal of Symbolic Logic, 71(2), pp. 677712.
Halbach, V. & Visser, A. (2014a). Self-reference in arithmetic I. The Review of Symbolic Logic, 7(04), 671691.
Halbach, V. & Visser, A. (2014b). Self-reference in arithmetic II. The Review of Symbolic Logic, 7(04), 692712.
Henkin, L. (1949). The completeness of the first-order functional calculus. Journal of Symbolic Logic, 14(03), 159166.
Hilbert, D. (1899). Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen. Leipzig: Teubner, pp. 192.
Hilbert, D. (1916). The Foundations of Physics: The Lectures (1916–1917). Reprinted in (Sauer & Majer, 2009).
Hilbert, D. (1917). Lectures on the principles of mathematics ‘prinzipien der mathematik’ (ws 1917/18). Reprinted in Ewald & Sieg (2013), pp. 31274.
Hilbert, D. (1922). Neubegründung der Mathematik: Erste Mitteilung. Abhandlungen aus dem Seminar der Hamburgischen Universität, 1, 157–77. English translation as “The new grounding of mathematics: First report” in Ewald (1996), pp. 1115–1134.
Hilbert, D. (1926). Über der Unendliche. Mathematische Annalen, 95, 161190. English translation as “On the infinite” in van Heijenoort (1967), pp. 292–367.
Hilbert, D. & Ackermann, W. (1928). Grundzüge der Theoretischen Logik (first edition). Berlin: Springer. Reprinted in Ewald & Sieg (2013).
Hilbert, D. & Ackermann, W. (1938). Grundzüge der Theoretischen Logik (second edition). Berlin: Springer. Translated as Hilbert & Ackermann (1950).
Hilbert, D. & Ackermann, W. (1950). Principles of Mathematical Logic. New York: Chelsea Publishing Company.
Hilbert, D. & Bernays, P. (1934). Grundlagen der Mathematik (second edition 1968), Vol. I. Berlin: Springer.
Hilbert, D. & Bernays, P. (1939). Grundlagen der Mathematik (second edition 1970), Vol. II. Berlin: Springer.
Horsten, L. (2011). The Tarskian Turn: Deflationism and Axiomatic Truth. Cambridge, MA: MIT Press.
Isaacson, D. (2011). The reality of mathematics and the case of set theory. In Novak, Z. and Simonyi, A., editors. Truth, Reference, and Realism. Budapest: Central European University Press, pp. 175.
Kanamori, A. (2009). Bernays and set theory. Bulletin of Symbolic Logic, 15(1), 4369.
Kaye, R. (1991). Models of Peano Arithmetic. Oxford Logic Guides, Vol. 15. Oxford: Oxford University Press.
Kaye, R. & Wong, T. (2007). On interpretations of arithmetic and set theory. Notre Dame Journal of Formal Logic, 48(4), 497510.
Kikuchi, M. (1997). Kolmogorov complexity and the second incompleteness theorem. Archive for Mathematical Logic, 36(6), 437443.
Kikuchi, M. & Kurahashi, T. (2016). Liar-type paradoxes and the incompleteness phenomena. Journal of Philosophical Logic, 45(4), 381398.
Kikuchi, M., Kurahashi, T., & Sakai, H. (2012). On proofs of the incompleteness theorems based on Berry’s paradox by Vopěnka, Chaitin, and Boolos. Mathematical Logic Quarterly, 58(4–5), 307316.
Kikuchi, M. & Tanaka, K. (1994). On formalization of model-theoretic proofs of Gödel’s theorems. Notre Dame Journal of Formal Logic, 35(3), 403412.
Kleene, S. (1943). Recursive predicates and quantifiers. Transactions of the American Mathematical Society, 53(1), 4173.
Kleene, S. (1952). Introduction to Metamathematics. Amsterdam: North-Holland.
Koellner, P. (2009). Truth in mathematics: The question of pluralism. In Linnebo, O. and Bueno, O., editors. New Waves in the Philosophy of Mathematics. New York: Palmgrave, pp. 80116.
Kotlarski, H. (2004). The incompleteness theorems after 70 years. Annals of Pure and Applied Logic, 126(1), 125138.
Kotlarski, H., Krajewski, S., & Lachlan, A. (1981). Construction of satisfaction classes for nonstandard models. Canadian Mathematical Bulletin, 14(3), 283293.
Kreisel, G. (1950). Note on arithmetic models for consistent formulae of the predicate calculus. Fundamenta Mathematicae, 37, 265285.
Kreisel, G. (1952). On the concepts of completeness and interpretation of formal systems. Fundamenta Mathematicae, 39, 103127.
Kreisel, G. (1953). Note on arithmetic models for consistent formulae of the predicate calculus. II. Actes du XIeme Congres International de Philosophie, Vol. XIV. Amsterdam: North-Holland, pp. 3949.
Kreisel, G. (1955). Models, translations and interpretations. In Skolem, T., editor. Mathematical Interpretation of Formal Systems. Amsterdam: North Holland, pp. 2650.
Kreisel, G. (1958). Wittgenstein’s remarks on the foundations of mathematics. The British Journal for the Philosophy of Science, 9(34), 135158.
Kreisel, G. (1965). Mathematical logic. In Saaty, T., editor. Lectures on Modern Mathematics, Vol. III. New York: Wiley, pp. 95195.
Kreisel, G. (1967). Informal rigour and completeness proofs. In Lakatos, I., editor. Problems in the Philosophy of Mathematics. Amsterdam: North-Holland, pp. 138186.
Kreisel, G. (1968). A survey of proof theory. The Journal of Symbolic Logic, 33(3), 321388.
Kreisel, G. (1969). Two notes on the foundations of set-theory. Dialectica, 23(2), 93114.
Kreisel, G. & Wang, H. (1955). Some applications of formalized consistency proofs. Fundamenta Mathematicae, 42, 101110.
Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690716.
Kripke, S. (2014). The road to Gödel. In Berg, J., editor. Naming, Necessity, and More. Berlin: Springer, pp. 223241.
Kritchman, S. & Raz, R. (2010). The surprise examination paradox and the second incompleteness theorem. Notices of the AMS, 57(11), 14541458.
Kruse, A. (1963). A method of modelling the formalism of set theory in axiomatic set theory. The Journal of Symbolic Logic, 28(1), 2034.
Lachlan, A. (1981). Full satisfaction and recursive saturation. Canadian Mathematical Bulletin, 24(3), 295297.
Lebesgue, H. (1905). Sur les fonctions représentables analytiquement. Journal de Mathematiques Pures et Appliquees, 1, 139216.
Lévy, A. (1976). The role of classes in set theory. Studies in Logic and the Foundations of Mathematics, 84, 173215.
Lindström, P. (1997). Aspects of Incompleteness. Lecture Notes in Logic, Vol. 10. Berlin: Springer.
Lusin, N. (1925). Sur les ensembles non mesurables B et l’emploi de la diagonale Cantor. Comptes rendus de l’Académie des Sciences Paris, 181, 9596.
Mancosu, P. (editor) (1998). From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford: Oxford University Press.
Mancosu, P. (2003). The Russellian influence on Hilbert and his school. Synthese, 137, 59101.
Manevitz, L. & Stavi, J. (1980). Operators and alternating sentences in arithmetic. Journal of Symbolic Logic, 45(01), 144154.
McGee, V. (1990). Truth, Vagueness and Paradox. Indianapolis: Hackett Publishers.
Mendelson, E. (1997). Introduction to Mathematical Logic (sixth edition). Boca Raton: CRC Press.
Montague, R. (1955). On the paradox of grounded classes. Journal of Symbolic Logic, 20(2), 140.
Mostowski, A. (1950). Some impredicative definitions in the axiomatic set-theory. Fundamenta Mathematicae, 38, 110124.
Müller, G. (1976). Sets and Classes: On the Work by Paul Bernays. Studies in Logic and the Foundations of Mathematics, Vol. 84. Amsterdam: North-Holland.
Myhill, J. (1952). The hypothesis that all classes are nameable. Proceedings of the National Academy of Sciences, 38(11), 979981.
Nelson, L. (1959). Beiträge zur Philosophie der Logik und Mathematik. Frankfurt am Main: Öffentliches Leben.
Nelson, L. & Grelling, K. (1908). Bemerkungen zu den Paradoxien von Russell und Burali-Forti. Abhandlungen der Fries’ schen Schule, Neue Folge, 2, 301334. Reprinted in Nelson (1959), pp. 57–77.
Novak, I. (1950). A construction for consistent systems. Fundamenta Mathematicae, 1(37), 87110.
Peckhaus, V. & Kahle, R. (2002). Hilbert’s paradox. Historia Mathematica, 29, 99.
Priest, G. (1994). The structure of the paradoxes of self-reference. Mind, 103(409), 2534.
Priest, G. (1997a). On a paradox of Hilbert and Bernays. Journal of Philosophical Logic, 26(1), 4556.
Priest, G. (1997b). Yablo’s paradox. Analysis, 57(4), 236242.
Putnam, H. (1957). Arithmetic models for consistent formulae of quantification theory. The Journal of Symbolic Logic, 22, 110111.
Putnam, H. (1965). Trial and error predicates and the solution to a problem of Mostowski. Journal of Symbolic Logic, 30(01), 4957.
Quine, W. (1981). Mathematical Logic. Cambridge, MA: Harvard University Press.
Rabin, M. (1958). On recursively enumerable and arithmetic models of set theory. Journal of Symbolic Logic, 23(4), 408416.
Ramsey, F. P. (1926). The foundations of mathematics. Proceedings of the London Mathematical Society, 2(1), 338384.
Read, S. (2016, August). Denotation, paradox and multiple meanings, manuscript.
Reinhardt, W. (1986). Some remarks on extending and interpreting theories with a partial predicate for truth. Journal of Philosophical Logic, 15(2), 219251.
Richard, J. (1905). The principles of mathematics and the problem of sets. Reprinted in van Heijenoort (1967), pp. 142144.
Robinson, A. (1963). On languages which are based on non-standard arithmetic. Nagoya Mathematical Journal, 22, 83117.
Rogers, H. (1987). Theory of Recursive Functions and Effective Computability. Cambridge, MA: MIT Press. First edition 1967.
Russell, B. (1903). The Principles of Mathematics. Cambridge: Cambridge University Press.
Russell, B. (1908). Mathematical logic as based on the theory of types. American Journal of Mathematics, 30(3), 222262.
Sauer, T. & Majer, U. (2009). David Hilbert’s Lectures on the Foundations of Physics 1915–1927: Relativity, Quantum Theory and Epistemology. Berlin: Springer.
Shoenfield, J. (1954). A relative consistency proof. The Journal of Symbolic Logic, 19, 2128.
Sieg, W. & Ravaglia, M. (2005). David Hilbert and Paul Bernays, Grundlagen der Mathematik, (1934, 1939). In Grattan-Guinness, I., editor. Landmark Writings in Western Mathematics 1640–1940. Amsterdam: Elsevier, p. 981.
Simpson, S. (2009). Subsystems of Second Order Arithmetic (second edition). Cambridge: Cambridge University Press.
Smorynski, C. (1977). The incompleteness theorems. In Barwise, J., editor. Handbook of Mathematical Logic. Amsterdam: North-Holland, pp. 821865.
Smorynski, C. (1984). Lectures on nonstandard models of arithmetic. In Lolli, G., Longo, G., and Marqa, A., editors. Logic Colloquium ’82. Amsterdam: North-Holland, pp. 170.
Smoryński, C. (1985). Self-Reference and Modal Logic. Amsterdam: Springer.
Tarski, A. (1935). Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica, 1, 261405. English translation as “The concept of truth in formalized languages” by J. H. Woodger in Tarski (1956).
Tarski, A. (1956). Logic, Semantics, Metamathematics—Papers from 1923 to 1938. Oxford: Clarendon Press.
Tarski, A., Mostowski, A., & Robinson, R. (1953). Undecidable Theories. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland.
van Heijenoort, J. (editor) (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press.
Visser, A. (1998). An overview of interpretability logic. In Kracht, M., de Rijke, M., and Wansing, H., editors. Advances in Modal Logic, Vol. 1. Stanford: CSLI Publications, pp. 307359.
Von Neumann, J. (1925). Eine axiomatisierung der mengenlehre. Journal für die reine und angewandte Mathematik, 154, 219240.
Vopĕnka, P. & Hájek, P. (1972). The Theory of Semisets. Amsterdam: North-Holland.
Vopĕnka, P. (1966). A new proof of Gödel’s results on non-provability of consistency. Bulletin de l’Académie Polonaise des Sciences, 14(3), 111.
Wang, H. (1953). Review: Note on arithmetic models for consistent formulae of the predicate calculus by G. Kreisel. Journal of Symbolic Logic, 18(2), 180181.
Wang, H. (1955). Undecidable sentences generated by semantic paradoxes. Journal of Symbolic Logic, 20(1), 3143.
Wang, H. (1963). A Survey of Mathematical Logic. Amsterdam: North Holland.
Wang, H. (1981). Popular Lectures on Mathematical Logic. Mineola: Dover.
Weyl, H. (1918). Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis. Leipzig: Verlag von Veit & Comp.
Weyl, H. (1919). Der circulus vitiosus in der heutigen Begründung der Analysis. Jahresbericht der Deutschen Mathematiker-Vereinigung, 28, 85102.
Zach, R. (1999). Completeness before Post: Bernays, Hilbert, and the development of propositional logic. Bulletin of Symbolic Logic, 5(03), 331366.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed