Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-25T15:10:40.682Z Has data issue: false hasContentIssue false
  • English
  • Français

INCOMPLETENESS VIA PARADOX AND COMPLETENESS

Published online by Cambridge University Press:  23 May 2019

WALTER DEAN
WALTER DEAN*
Affiliation:
Department of Philosophy, University of Warwick
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF WARWICK COVENTRY CV4 7AL, UK E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

03C6203H1503C5700A30Arithmetized Completeness Theoreminterpretabilityset theoretic paradoxessemantic paradoxesHilbert programDavid HilbertPaul BernaysGeorg KreiselHao Wang
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 3 , September 2020 , pp. 541 - 592
Check for updates
Copyright
Copyright © Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

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.Google Scholar
Ackermann, W. (1928). Über die Erfüllbarkeit gewisser Zählausdrücke. Mathematische Annalen, 100 (1), 638–649.10.1007/BF01448869CrossRefGoogle Scholar
Bernays, P. (1930). Die Philosophie der Mathematik und die Hilbertsche Beweistheorie. Blätter für deutsche Philosophie, 4, 326–367. Reprinted in Bernays (1976), pp. 17–62 and in Mancosu (1998), pp. 234–265.Google Scholar
Bernays, P. (1937). A system of axiomatic set theory: Part I. Journal of Symbolic Logic, 2(1), 65–77.Google Scholar
Bernays, P. (1942). A system of axiomatic set theory: Part III. Infinity and enumerability. Analysis. Journal of Symbolic Logic, 7(2), 65–89.10.2307/2266303CrossRefGoogle Scholar
Bernays, P. (1954a). A system of axiomatic set theory: Part VII. Journal of Symbolic Logic, 19(2), 81–96.10.2307/2268864CrossRefGoogle Scholar
Bernays, P. (1954b). Zur Beurteilung der situation in der beweistheoretischen Forschung. Revue Internationale de Philosophie, 27/28, 9–13.Google Scholar
Bernays, P. (1970). Die schematische Korrespondenz und die idealisierten Strukturen. Dialectica, 24, 53–66. Reprinted in Bernays (1976), pp. 176–188.10.1111/j.1746-8361.1970.tb01203.xCrossRefGoogle Scholar
Bernays, P. (1976). Abhandlungen zur Philosophie der Mathematik. Darmstadt: Wiss. Buchgesellschaft.Google Scholar
Bernays, P. & Fraenkel, A. (1958). Axiomatic Set Theory. Amsterdam: North-Holland.Google Scholar
Boolos, G. (1989). A new proof of the Gödel incompleteness theorem. Notices of the American Mathematical Society, 36(4), 388–390.Google Scholar
Borel, É. (1898). Leçons sur la théorie des fonctions. Paris: Gauthier-Villars et fils.Google Scholar
Church, A. (1956). Introduction to Mathematical Logic. Princeton: Princeton University Press.Google Scholar
Church, A. (1976). Comparison of Russell’s resolution of the semantical antinomies with that of Tarski. The Journal of Symbolic Logic, 41(4), 747–760.10.2307/2272393CrossRefGoogle Scholar
Cieśliński, C. (2002). Heterologicality and incompleteness. Mathematical Logic Quarterly, 48(1), 105–110.10.1002/1521-3870(200201)48:1<105::AID-MALQ105>3.0.CO;2-V3.0.CO;2-V>CrossRefGoogle Scholar
Cieśliński, C. (2018). The Epistemic Lightness of Truth: Deflationism and its Logic. Cambridge: Cambridge University Press.Google Scholar
Cohen, P. (1966). Set Theory and the Continuum Hypothesis. New York: W.A. Benjamin.Google Scholar
Dean, W. (2015). Arithmetical reflection and the provability of soundness. Philosophia Mathematica, 23(1), 31–64.10.1093/philmat/nku026CrossRefGoogle Scholar
Dean, W. (2017). Bernays and the completeness theorem. Annals of the Japanese Association for the Philosophy of Science, 25, 44–55.10.4288/jafpos.25.0_45CrossRefGoogle Scholar
Dean, W. & Walsh, S. (2017). The prehistory of the subsystems of second-order arithmetic. The Review of Symbolic Logic, 10(2), 357–396.10.1017/S1755020316000411CrossRefGoogle Scholar
Doets, K. (1999). Relatives of the Russell paradox. Mathematical Logic Quarterly, 45, 73–83.10.1002/malq.19990450107CrossRefGoogle Scholar
Dummett, M. (1978). Frege’s distinction between sense and reference. Truth and Other Enigmas. Cambridge, MA: Harvard University Press, pp. 116–144.Google Scholar
Ebbs, G. (2015). Satisfying predicates: Kleene’s proof of the Hilbert–Bernays theorem. History and Philosophy of Logic, 36(4), 346–366.CrossRefGoogle Scholar
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. 321–335.10.1007/978-94-017-9673-6_16CrossRefGoogle Scholar
Enayat, A., Łełyk, M., & Wcisło, B. (2019). Truth and feasible reducibility. The Journal of Symbolic Logic, to appear.Google Scholar
Ewald, W. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press.Google Scholar
Ewald, W. & Sieg, W. (editors) (2013). David Hilbert’s Lectures on the Foundations of Logic and Arithmetic 1917–1933. Berlin: Springer.CrossRefGoogle Scholar
Feferman, S. (1960). Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49, 35–92.CrossRefGoogle Scholar
Feferman, S. (1991). Reflecting on incompleteness. Journal of Symbolic Logic, 56(1), 1–49.10.2307/2274902CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Field, H. (2008). Saving Truth from Paradox. Oxford: Oxford University Press.CrossRefGoogle Scholar
Fischer, M., Horsten, L., & Nicolai, C. (2019). Hypatia’s silence: Truth, justification, and entitlement. Noûs, to appear.10.1111/nous.12292CrossRefGoogle Scholar
Fitch, F. (1952). Symbolic Logic. New York: The Ronald Press Company.Google Scholar
Gentzen, G. (1936). Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, 112, 493–565.CrossRefGoogle Scholar
Gödel, K. (1930). The completeness of the axioms of the functional calculus of logic. pp. 103–123. Reprinted in Feferman et al. (1986).Google Scholar
Gödel, K. (1931a). Correspondence with Ernest Zermelo. Reprinted in Feferman et al. . (2003).Google Scholar
Gödel, K. (1931b). On formally undecidable propositions of Principia Mathematica and related systems I. Reprinted in Feferman et al. (1986).Google Scholar
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.Google Scholar
Hájek, P. & Pudlák, P. (1998). Metamathematics of First-Order Arithmetic (first edition 1993). Berlin: Springer.Google Scholar
Halbach, V. (2011). Axiomatic Theories of Truth. Cambridge: Cambridge University Press.10.1017/CBO9780511921049CrossRefGoogle Scholar
Halbach, V. & Horsten, L. (2006). Axiomatizing Kripke’s theory of truth. The Journal of Symbolic Logic, 71(2), pp. 677–712.CrossRefGoogle Scholar
Halbach, V. & Visser, A. (2014a). Self-reference in arithmetic I. The Review of Symbolic Logic, 7(04), 671–691.CrossRefGoogle Scholar
Halbach, V. & Visser, A. (2014b). Self-reference in arithmetic II. The Review of Symbolic Logic, 7(04), 692–712.CrossRefGoogle Scholar
Henkin, L. (1949). The completeness of the first-order functional calculus. Journal of Symbolic Logic, 14(03), 159–166.CrossRefGoogle Scholar
Hilbert, D. (1899). Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen. Leipzig: Teubner, pp. 1–92.Google Scholar
Hilbert, D. (1916). The Foundations of Physics: The Lectures (1916–1917). Reprinted in (Sauer & Majer, 2009).Google Scholar
Hilbert, D. (1917). Lectures on the principles of mathematics ‘prinzipien der mathematik’ (ws 1917/18). Reprinted in Ewald & Sieg (2013), pp. 31–274.Google Scholar
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.Google Scholar
Hilbert, D. (1926). Über der Unendliche. Mathematische Annalen, 95, 161–190. English translation as “On the infinite” in van Heijenoort (1967), pp. 292–367.Google Scholar
Hilbert, D. & Ackermann, W. (1928). Grundzüge der Theoretischen Logik (first edition). Berlin: Springer. Reprinted in Ewald & Sieg (2013).Google Scholar
Hilbert, D. & Ackermann, W. (1938). Grundzüge der Theoretischen Logik (second edition). Berlin: Springer. Translated as Hilbert & Ackermann (1950).CrossRefGoogle Scholar
Hilbert, D. & Ackermann, W. (1950). Principles of Mathematical Logic. New York: Chelsea Publishing Company.Google Scholar
Hilbert, D. & Bernays, P. (1934). Grundlagen der Mathematik (second edition 1968), Vol. I. Berlin: Springer.Google Scholar
Hilbert, D. & Bernays, P. (1939). Grundlagen der Mathematik (second edition 1970), Vol. II. Berlin: Springer.Google Scholar
Horsten, L. (2011). The Tarskian Turn: Deflationism and Axiomatic Truth. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
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. 1–75.Google Scholar
Kanamori, A. (2009). Bernays and set theory. Bulletin of Symbolic Logic, 15(1), 43–69.CrossRefGoogle Scholar
Kaye, R. (1991). Models of Peano Arithmetic. Oxford Logic Guides, Vol. 15. Oxford: Oxford University Press.Google Scholar
Kaye, R. & Wong, T. (2007). On interpretations of arithmetic and set theory. Notre Dame Journal of Formal Logic, 48(4), 497–510.CrossRefGoogle Scholar
Kikuchi, M. (1997). Kolmogorov complexity and the second incompleteness theorem. Archive for Mathematical Logic, 36(6), 437–443.CrossRefGoogle Scholar
Kikuchi, M. & Kurahashi, T. (2016). Liar-type paradoxes and the incompleteness phenomena. Journal of Philosophical Logic, 45(4), 381–398.CrossRefGoogle Scholar
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), 307–316.CrossRefGoogle Scholar
Kikuchi, M. & Tanaka, K. (1994). On formalization of model-theoretic proofs of Gödel’s theorems. Notre Dame Journal of Formal Logic, 35(3), 403–412.10.1305/ndjfl/1040511346CrossRefGoogle Scholar
Kleene, S. (1943). Recursive predicates and quantifiers. Transactions of the American Mathematical Society, 53(1), 41–73.10.1090/S0002-9947-1943-0007371-8CrossRefGoogle Scholar
Kleene, S. (1952). Introduction to Metamathematics. Amsterdam: North-Holland.Google Scholar
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. 80–116.CrossRefGoogle Scholar
Kotlarski, H. (2004). The incompleteness theorems after 70 years. Annals of Pure and Applied Logic, 126(1), 125–138.CrossRefGoogle Scholar
Kotlarski, H., Krajewski, S., & Lachlan, A. (1981). Construction of satisfaction classes for nonstandard models. Canadian Mathematical Bulletin, 14(3), 283–293.CrossRefGoogle Scholar
Kreisel, G. (1950). Note on arithmetic models for consistent formulae of the predicate calculus. Fundamenta Mathematicae, 37, 265–285.10.4064/fm-37-1-265-285CrossRefGoogle Scholar
Kreisel, G. (1952). On the concepts of completeness and interpretation of formal systems. Fundamenta Mathematicae, 39, 103–127.CrossRefGoogle Scholar
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. 39–49.Google Scholar
Kreisel, G. (1955). Models, translations and interpretations. In Skolem, T., editor. Mathematical Interpretation of Formal Systems. Amsterdam: North Holland, pp. 26–50.CrossRefGoogle Scholar
Kreisel, G. (1958). Wittgenstein’s remarks on the foundations of mathematics. The British Journal for the Philosophy of Science, 9(34), 135–158.CrossRefGoogle Scholar
Kreisel, G. (1965). Mathematical logic. In Saaty, T., editor. Lectures on Modern Mathematics, Vol. III. New York: Wiley, pp. 95–195.Google Scholar
Kreisel, G. (1967). Informal rigour and completeness proofs. In Lakatos, I., editor. Problems in the Philosophy of Mathematics. Amsterdam: North-Holland, pp. 138–186.10.1016/S0049-237X(08)71525-8CrossRefGoogle Scholar
Kreisel, G. (1968). A survey of proof theory. The Journal of Symbolic Logic, 33(3), 321–388.10.2307/2270324CrossRefGoogle Scholar
Kreisel, G. (1969). Two notes on the foundations of set-theory. Dialectica, 23(2), 93–114.CrossRefGoogle Scholar
Kreisel, G. & Wang, H. (1955). Some applications of formalized consistency proofs. Fundamenta Mathematicae, 42, 101–110.CrossRefGoogle Scholar
Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–716.CrossRefGoogle Scholar
Kripke, S. (2014). The road to Gödel. In Berg, J., editor. Naming, Necessity, and More. Berlin: Springer, pp. 223–241.Google Scholar
Kritchman, S. & Raz, R. (2010). The surprise examination paradox and the second incompleteness theorem. Notices of the AMS, 57(11), 1454–1458.Google Scholar
Kruse, A. (1963). A method of modelling the formalism of set theory in axiomatic set theory. The Journal of Symbolic Logic, 28(1), 20–34.CrossRefGoogle Scholar
Lachlan, A. (1981). Full satisfaction and recursive saturation. Canadian Mathematical Bulletin, 24(3), 295–297.CrossRefGoogle Scholar
Lebesgue, H. (1905). Sur les fonctions représentables analytiquement. Journal de Mathematiques Pures et Appliquees, 1, 139–216.Google Scholar
Lévy, A. (1976). The role of classes in set theory. Studies in Logic and the Foundations of Mathematics, 84, 173–215.CrossRefGoogle Scholar
Lindström, P. (1997). Aspects of Incompleteness. Lecture Notes in Logic, Vol. 10. Berlin: Springer.CrossRefGoogle Scholar
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, 95–96.Google Scholar
Mancosu, P. (editor) (1998). From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford: Oxford University Press.Google Scholar
Mancosu, P. (2003). The Russellian influence on Hilbert and his school. Synthese, 137, 59–101.CrossRefGoogle Scholar
Manevitz, L. & Stavi, J. (1980). Operators and alternating sentences in arithmetic. Journal of Symbolic Logic, 45(01), 144–154.CrossRefGoogle Scholar
McGee, V. (1990). Truth, Vagueness and Paradox. Indianapolis: Hackett Publishers.Google Scholar
Mendelson, E. (1997). Introduction to Mathematical Logic (sixth edition). Boca Raton: CRC Press.Google Scholar
Montague, R. (1955). On the paradox of grounded classes. Journal of Symbolic Logic, 20(2), 140.CrossRefGoogle Scholar
Mostowski, A. (1950). Some impredicative definitions in the axiomatic set-theory. Fundamenta Mathematicae, 38, 110–124.Google Scholar
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.Google Scholar
Myhill, J. (1952). The hypothesis that all classes are nameable. Proceedings of the National Academy of Sciences, 38(11), 979–981.CrossRefGoogle ScholarPubMed
Nelson, L. (1959). Beiträge zur Philosophie der Logik und Mathematik. Frankfurt am Main: Öffentliches Leben.Google Scholar
Nelson, L. & Grelling, K. (1908). Bemerkungen zu den Paradoxien von Russell und Burali-Forti. Abhandlungen der Fries’ schen Schule, Neue Folge, 2, 301–334. Reprinted in Nelson (1959), pp. 57–77.Google Scholar
Novak, I. (1950). A construction for consistent systems. Fundamenta Mathematicae, 1(37), 87–110.CrossRefGoogle Scholar
Peckhaus, V. & Kahle, R. (2002). Hilbert’s paradox. Historia Mathematica, 29, 99.CrossRefGoogle Scholar
Priest, G. (1994). The structure of the paradoxes of self-reference. Mind, 103(409), 25–34.CrossRefGoogle Scholar
Priest, G. (1997a). On a paradox of Hilbert and Bernays. Journal of Philosophical Logic, 26(1), 45–56.CrossRefGoogle Scholar
Priest, G. (1997b). Yablo’s paradox. Analysis, 57(4), 236–242.CrossRefGoogle Scholar
Putnam, H. (1957). Arithmetic models for consistent formulae of quantification theory. The Journal of Symbolic Logic, 22, 110–111.Google Scholar
Putnam, H. (1965). Trial and error predicates and the solution to a problem of Mostowski. Journal of Symbolic Logic, 30(01), 49–57.CrossRefGoogle Scholar
Quine, W. (1981). Mathematical Logic. Cambridge, MA: Harvard University Press.Google Scholar
Rabin, M. (1958). On recursively enumerable and arithmetic models of set theory. Journal of Symbolic Logic, 23(4), 408–416.10.2307/2964015CrossRefGoogle Scholar
Ramsey, F. P. (1926). The foundations of mathematics. Proceedings of the London Mathematical Society, 2(1), 338–384.CrossRefGoogle Scholar
Read, S. (2016, August). Denotation, paradox and multiple meanings, manuscript.Google Scholar
Reinhardt, W. (1986). Some remarks on extending and interpreting theories with a partial predicate for truth. Journal of Philosophical Logic, 15(2), 219–251.CrossRefGoogle Scholar
Richard, J. (1905). The principles of mathematics and the problem of sets. Reprinted in van Heijenoort (1967), pp. 142–144.Google Scholar
Robinson, A. (1963). On languages which are based on non-standard arithmetic. Nagoya Mathematical Journal, 22, 83–117.CrossRefGoogle Scholar
Rogers, H. (1987). Theory of Recursive Functions and Effective Computability. Cambridge, MA: MIT Press. First edition 1967.Google Scholar
Russell, B. (1903). The Principles of Mathematics. Cambridge: Cambridge University Press.Google Scholar
Russell, B. (1908). Mathematical logic as based on the theory of types. American Journal of Mathematics, 30(3), 222–262.CrossRefGoogle Scholar
Sauer, T. & Majer, U. (2009). David Hilbert’s Lectures on the Foundations of Physics 1915–1927: Relativity, Quantum Theory and Epistemology. Berlin: Springer.10.1007/b12915CrossRefGoogle Scholar
Shoenfield, J. (1954). A relative consistency proof. The Journal of Symbolic Logic, 19, 21–28.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Simpson, S. (2009). Subsystems of Second Order Arithmetic (second edition). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Smorynski, C. (1977). The incompleteness theorems. In Barwise, J., editor. Handbook of Mathematical Logic. Amsterdam: North-Holland, pp. 821–865.CrossRefGoogle Scholar
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. 1–70.Google Scholar
Smoryński, C. (1985). Self-Reference and Modal Logic. Amsterdam: Springer.CrossRefGoogle Scholar
Tarski, A. (1935). Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica, 1, 261–405. English translation as “The concept of truth in formalized languages” by J. H. Woodger in Tarski (1956).Google Scholar
Tarski, A. (1956). Logic, Semantics, Metamathematics—Papers from 1923 to 1938. Oxford: Clarendon Press.Google Scholar
Tarski, A., Mostowski, A., & Robinson, R. (1953). Undecidable Theories. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland.Google Scholar
van Heijenoort, J. (editor) (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press.Google Scholar
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. 307–359.Google Scholar
Von Neumann, J. (1925). Eine axiomatisierung der mengenlehre. Journal für die reine und angewandte Mathematik, 154, 219–240.Google Scholar
Vopĕnka, P. & Hájek, P. (1972). The Theory of Semisets. Amsterdam: North-Holland.Google Scholar
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.Google Scholar
Wang, H. (1953). Review: Note on arithmetic models for consistent formulae of the predicate calculus by G. Kreisel. Journal of Symbolic Logic, 18(2), 180–181.CrossRefGoogle Scholar
Wang, H. (1955). Undecidable sentences generated by semantic paradoxes. Journal of Symbolic Logic, 20(1), 31–43.CrossRefGoogle Scholar
Wang, H. (1963). A Survey of Mathematical Logic. Amsterdam: North Holland.Google Scholar
Wang, H. (1981). Popular Lectures on Mathematical Logic. Mineola: Dover.Google Scholar
Weyl, H. (1918). Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis. Leipzig: Verlag von Veit & Comp.Google Scholar
Weyl, H. (1919). Der circulus vitiosus in der heutigen Begründung der Analysis. Jahresbericht der Deutschen Mathematiker-Vereinigung, 28, 85–102.Google Scholar
Zach, R. (1999). Completeness before Post: Bernays, Hilbert, and the development of propositional logic. Bulletin of Symbolic Logic, 5(03), 331–366.CrossRefGoogle Scholar