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Incompleteness via paradox and completeness. (English) Zbl 1485.03244

Summary: 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.

MSC:

03F40 Gödel numberings and issues of incompleteness
03F30 First-order arithmetic and fragments
03C62 Models of arithmetic and set theory
03H15 Nonstandard models of arithmetic
03A05 Philosophical and critical aspects of logic and foundations
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[1] , 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.
[2] Ackermann, W. (1928). Über die Erfüllbarkeit gewisser Zählausdrücke. Mathematische Annalen, 100 (1), 638-649. · JFM 54.0057.01
[3] 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. · JFM 56.0044.02
[4] Bernays, P. (1937). A system of axiomatic set theory: Part I. Journal of Symbolic Logic, 2(1), 65-77. · JFM 63.0028.01
[5] Bernays, P. (1942). A system of axiomatic set theory: Part III. Infinity and enumerability. Analysis. Journal of Symbolic Logic, 7(2), 65-89. · Zbl 0061.09201
[6] Bernays, P. (1954a). A system of axiomatic set theory: Part VII. Journal of Symbolic Logic, 19(2), 81-96. · Zbl 0055.04603
[7] Bernays, P. (1954b). Zur Beurteilung der situation in der beweistheoretischen Forschung. Revue Internationale de Philosophie, 27/28, 9-13.
[8] Bernays, P. (1970). Die schematische Korrespondenz und die idealisierten Strukturen. Dialectica, 24, 53-66. Reprinted in Bernays (1976), pp. 176-188. · Zbl 0257.02003
[9] Bernays, P. (1976). Abhandlungen zur Philosophie der Mathematik. Darmstadt: Wiss. Buchgesellschaft. · Zbl 0335.02002
[10] Bernays, P. & Fraenkel, A. (1958). Axiomatic Set Theory. Amsterdam: North-Holland. · Zbl 0175.27002
[11] Boolos, G. (1989). A new proof of the Gödel incompleteness theorem. Notices of the American Mathematical Society, 36(4), 388-390. · Zbl 0972.03544
[12] Borel, É. (1898). Leçons sur la théorie des fonctions. Paris: Gauthier-Villars et fils. · JFM 29.0336.01
[13] Church, A. (1956). Introduction to Mathematical Logic. Princeton: Princeton University Press. · Zbl 0073.24301
[14] 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. · Zbl 0383.03005
[15] Cieśliński, C. (2002). Heterologicality and incompleteness. Mathematical Logic Quarterly, 48(1), 105-110. · Zbl 0990.03028
[16] Cieśliński, C. (2018). The Epistemic Lightness of Truth: Deflationism and its Logic. Cambridge: Cambridge University Press.
[17] Cohen, P. (1966). Set Theory and the Continuum Hypothesis. New York: W.A. Benjamin. · Zbl 0182.01301
[18] Dean, W. (2015). Arithmetical reflection and the provability of soundness. Philosophia Mathematica, 23(1), 31-64. · Zbl 1380.03068
[19] Dean, W. (2017). Bernays and the completeness theorem. Annals of the Japanese Association for the Philosophy of Science, 25, 44-55. · Zbl 1506.03060
[20] Dean, W. & Walsh, S. (2017). The prehistory of the subsystems of second-order arithmetic. The Review of Symbolic Logic, 10(2), 357-396. · Zbl 1376.03005
[21] Doets, K. (1999). Relatives of the Russell paradox. Mathematical Logic Quarterly, 45, 73-83. · Zbl 0924.03095
[22] Dummett, M. (1978). Frege’s distinction between sense and reference. Truth and Other Enigmas. Cambridge, MA: Harvard University Press, pp. 116-144.
[23] Ebbs, G. (2015). Satisfying predicates: Kleene’s proof of the Hilbert-Bernays theorem. History and Philosophy of Logic, 36(4), 346-366. · Zbl 1369.03004
[24] 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.
[25] Enayat, A., Łełyk, M., & Wcisło, B. (2019). Truth and feasible reducibility. The Journal of Symbolic Logic, to appear. · Zbl 1444.03164
[26] Ewald, W. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. · Zbl 0859.01002
[27] Ewald, W. & Sieg, W. (editors) (2013). David Hilbert’s Lectures on the Foundations of Logic and Arithmetic 1917-1933. Berlin: Springer. · Zbl 1275.03002
[28] Feferman, S. (1960). Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49, 35-92. · Zbl 0095.24301
[29] Feferman, S. (1991). Reflecting on incompleteness. Journal of Symbolic Logic, 56(1), 1-49. · Zbl 0746.03046
[30] 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.
[31] 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. · Zbl 1026.01019
[32] Field, H. (2008). Saving Truth from Paradox. Oxford: Oxford University Press. · Zbl 1225.03006
[33] Fischer, M., Horsten, L., & Nicolai, C. (2019). Hypatia’s silence: Truth, justification, and entitlement. Noûs, to appear.
[34] Fitch, F. (1952). Symbolic Logic. New York: The Ronald Press Company. · Zbl 0049.00504
[35] Gentzen, G. (1936). Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, 112, 493-565. · JFM 62.0044.01
[36] Gödel, K. (1930). The completeness of the axioms of the functional calculus of logic. pp. 103-123. Reprinted in Feferman et al. (1986).
[37] Gödel, K. (1931a). Correspondence with Ernest Zermelo. Reprinted in Feferman et al.. (2003).
[38] Gödel, K. (1931b). On formally undecidable propositions of Principia Mathematica and related systems I. Reprinted in Feferman et al. (1986). · Zbl 0002.00101
[39] 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. · Zbl 0061.00902
[40] Hájek, P. & Pudlák, P. (1998). Metamathematics of First-Order Arithmetic (first edition 1993). Berlin: Springer. · Zbl 0781.03047
[41] Halbach, V. (2011). Axiomatic Theories of Truth. Cambridge: Cambridge University Press. · Zbl 1223.03001
[42] Halbach, V. & Horsten, L. (2006). Axiomatizing Kripke’s theory of truth. The Journal of Symbolic Logic, 71(2), pp. 677-712. · Zbl 1101.03005
[43] Halbach, V. & Visser, A. (2014a). Self-reference in arithmetic I. The Review of Symbolic Logic, 7(04), 671-691. · Zbl 1337.03008
[44] Halbach, V. & Visser, A. (2014b). Self-reference in arithmetic II. The Review of Symbolic Logic, 7(04), 692-712. · Zbl 1337.03009
[45] Henkin, L. (1949). The completeness of the first-order functional calculus. Journal of Symbolic Logic, 14(03), 159-166. · Zbl 0034.00602
[46] Hilbert, D. (1899). Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen. Leipzig: Teubner, pp. 1-92. · JFM 30.0424.01
[47] Hilbert, D. (1916). The Foundations of Physics: The Lectures (1916-1917). Reprinted in (Sauer & Majer, 2009).
[48] Hilbert, D. (1917). Lectures on the principles of mathematics ‘prinzipien der mathematik’ (ws 1917/18). Reprinted in Ewald & Sieg (2013), pp. 31-274.
[49] 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.
[50] Hilbert, D. (1926). Über der Unendliche. Mathematische Annalen, 95, 161-190. English translation as “On the infinite” in van Heijenoort (1967), pp. 292-367.
[51] Hilbert, D. & Ackermann, W. (1928). Grundzüge der Theoretischen Logik (first edition). Berlin: Springer. Reprinted in Ewald & Sieg (2013). · JFM 54.0055.01
[52] Hilbert, D. & Ackermann, W. (1938). Grundzüge der Theoretischen Logik (second edition). Berlin: Springer. Translated as Hilbert & Ackermann (1950). · JFM 64.0026.05
[53] Hilbert, D. & Ackermann, W. (1950). Principles of Mathematical Logic. New York: Chelsea Publishing Company. · Zbl 0040.00402
[54] Hilbert, D. & Bernays, P. (1934). Grundlagen der Mathematik (second edition 1968), Vol. I. Berlin: Springer. · JFM 60.0017.02
[55] Hilbert, D. & Bernays, P. (1939). Grundlagen der Mathematik (second edition 1970), Vol. II. Berlin: Springer. · Zbl 0020.19301
[56] Horsten, L. (2011). The Tarskian Turn: Deflationism and Axiomatic Truth. Cambridge, MA: MIT Press. · Zbl 1242.03001
[57] 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.
[58] Kanamori, A. (2009). Bernays and set theory. Bulletin of Symbolic Logic, 15(1), 43-69. · Zbl 1172.03002
[59] Kaye, R. (1991). Models of Peano Arithmetic. Oxford Logic Guides, Vol. 15. Oxford: Oxford University Press. · Zbl 0744.03037
[60] Kaye, R. & Wong, T. (2007). On interpretations of arithmetic and set theory. Notre Dame Journal of Formal Logic, 48(4), 497-510. · Zbl 1137.03019
[61] Kikuchi, M. (1997). Kolmogorov complexity and the second incompleteness theorem. Archive for Mathematical Logic, 36(6), 437-443. · Zbl 0883.03042
[62] Kikuchi, M. & Kurahashi, T. (2016). Liar-type paradoxes and the incompleteness phenomena. Journal of Philosophical Logic, 45(4), 381-398. · Zbl 1350.03019
[63] 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. · Zbl 1257.03088
[64] 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. · Zbl 0822.03032
[65] Kleene, S. (1943). Recursive predicates and quantifiers. Transactions of the American Mathematical Society, 53(1), 41-73. · Zbl 0063.03259
[66] Kleene, S. (1952). Introduction to Metamathematics. Amsterdam: North-Holland. · Zbl 0047.00703
[67] 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. · Zbl 1343.00010
[68] Kotlarski, H. (2004). The incompleteness theorems after 70 years. Annals of Pure and Applied Logic, 126(1), 125-138. · Zbl 1053.03033
[69] Kotlarski, H., Krajewski, S., & Lachlan, A. (1981). Construction of satisfaction classes for nonstandard models. Canadian Mathematical Bulletin, 14(3), 283-293. · Zbl 0471.03054
[70] Kreisel, G. (1950). Note on arithmetic models for consistent formulae of the predicate calculus. Fundamenta Mathematicae, 37, 265-285. · Zbl 0040.00302
[71] Kreisel, G. (1952). On the concepts of completeness and interpretation of formal systems. Fundamenta Mathematicae, 39, 103-127. · Zbl 0050.00601
[72] 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. · Zbl 0053.20004
[73] Kreisel, G. (1955). Models, translations and interpretations. In Skolem, T., editor. Mathematical Interpretation of Formal Systems. Amsterdam: North Holland, pp. 26-50. · Zbl 0066.00902
[74] Kreisel, G. (1958). Wittgenstein’s remarks on the foundations of mathematics. The British Journal for the Philosophy of Science, 9(34), 135-158.
[75] Kreisel, G. (1965). Mathematical logic. In Saaty, T., editor. Lectures on Modern Mathematics, Vol. III. New York: Wiley, pp. 95-195. · Zbl 0147.24703
[76] Kreisel, G. (1967). Informal rigour and completeness proofs. In Lakatos, I., editor. Problems in the Philosophy of Mathematics. Amsterdam: North-Holland, pp. 138-186.
[77] Kreisel, G. (1968). A survey of proof theory. The Journal of Symbolic Logic, 33(3), 321-388. · Zbl 0177.01002
[78] Kreisel, G. (1969). Two notes on the foundations of set-theory. Dialectica, 23(2), 93-114. · Zbl 0255.02002
[79] Kreisel, G. & Wang, H. (1955). Some applications of formalized consistency proofs. Fundamenta Mathematicae, 42, 101-110. · Zbl 0067.25201
[80] Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690-716. · Zbl 0952.03513
[81] Kripke, S. (2014). The road to Gödel. In Berg, J., editor. Naming, Necessity, and More. Berlin: Springer, pp. 223-241.
[82] Kritchman, S. & Raz, R. (2010). The surprise examination paradox and the second incompleteness theorem. Notices of the AMS, 57(11), 1454-1458. · Zbl 1261.03159
[83] 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. · Zbl 0233.02027
[84] Lachlan, A. (1981). Full satisfaction and recursive saturation. Canadian Mathematical Bulletin, 24(3), 295-297. · Zbl 0471.03055
[85] Lebesgue, H. (1905). Sur les fonctions représentables analytiquement. Journal de Mathematiques Pures et Appliquees, 1, 139-216. · JFM 36.0453.02
[86] Lévy, A. (1976). The role of classes in set theory. Studies in Logic and the Foundations of Mathematics, 84, 173-215. · Zbl 0336.02046
[87] Lindström, P. (1997). Aspects of Incompleteness. Lecture Notes in Logic, Vol. 10. Berlin: Springer. · Zbl 0882.03054
[88] 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. · JFM 51.0169.04
[89] Mancosu, P. (editor) (1998). From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford: Oxford University Press. · Zbl 0941.01003
[90] Mancosu, P. (2003). The Russellian influence on Hilbert and his school. Synthese, 137, 59-101.
[91] Manevitz, L. & Stavi, J. (1980). Operators and alternating sentences in arithmetic. Journal of Symbolic Logic, 45(01), 144-154. · Zbl 0458.03022
[92] Mcgee, V. (1990). Truth, Vagueness and Paradox. Indianapolis: Hackett Publishers. · Zbl 0734.03001
[93] Mendelson, E. (1997). Introduction to Mathematical Logic (sixth edition). Boca Raton: CRC Press. · Zbl 0915.03002
[94] Montague, R. (1955). On the paradox of grounded classes. Journal of Symbolic Logic, 20(2), 140. · Zbl 0068.24702
[95] Mostowski, A. (1950). Some impredicative definitions in the axiomatic set-theory. Fundamenta Mathematicae, 38, 110-124. · Zbl 0039.27601
[96] 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. · Zbl 0327.00003
[97] Myhill, J. (1952). The hypothesis that all classes are nameable. Proceedings of the National Academy of Sciences, 38(11), 979-981. · Zbl 0049.14903
[98] Nelson, L. (1959). Beiträge zur Philosophie der Logik und Mathematik. Frankfurt am Main: Öffentliches Leben. · Zbl 0087.00712
[99] 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.
[100] Novak, I. (1950). A construction for consistent systems. Fundamenta Mathematicae, 1(37), 87-110. · Zbl 0039.24504
[101] Peckhaus, V. & Kahle, R. (2002). Hilbert’s paradox. Historia Mathematica, 29, 99. · Zbl 0996.01012
[102] Priest, G. (1994). The structure of the paradoxes of self-reference. Mind, 103(409), 25-34.
[103] Priest, G. (1997a). On a paradox of Hilbert and Bernays. Journal of Philosophical Logic, 26(1), 45-56. · Zbl 0869.03005
[104] Priest, G. (1997b). Yablo’s paradox. Analysis, 57(4), 236-242. · Zbl 0943.03588
[105] Putnam, H. (1957). Arithmetic models for consistent formulae of quantification theory. The Journal of Symbolic Logic, 22, 110-111.
[106] Putnam, H. (1965). Trial and error predicates and the solution to a problem of Mostowski. Journal of Symbolic Logic, 30(01), 49-57. · Zbl 0193.30102
[107] Quine, W. (1981). Mathematical Logic. Cambridge, MA: Harvard University Press. · Zbl 0579.03001
[108] Rabin, M. (1958). On recursively enumerable and arithmetic models of set theory. Journal of Symbolic Logic, 23(4), 408-416. · Zbl 0095.24601
[109] Ramsey, F. P. (1926). The foundations of mathematics. Proceedings of the London Mathematical Society, 2(1), 338-384. · JFM 52.0046.01
[110] Read, S. (2016, August). Denotation, paradox and multiple meanings, manuscript.
[111] Reinhardt, W. (1986). Some remarks on extending and interpreting theories with a partial predicate for truth. Journal of Philosophical Logic, 15(2), 219-251. · Zbl 0629.03002
[112] Richard, J. (1905). The principles of mathematics and the problem of sets. Reprinted in van Heijenoort (1967), pp. 142-144.
[113] Robinson, A. (1963). On languages which are based on non-standard arithmetic. Nagoya Mathematical Journal, 22, 83-117. · Zbl 0166.26101
[114] Rogers, H. (1987). Theory of Recursive Functions and Effective Computability. Cambridge, MA: MIT Press. First edition 1967. · Zbl 0183.01401
[115] Russell, B. (1903). The Principles of Mathematics. Cambridge: Cambridge University Press. · JFM 34.0062.14
[116] Russell, B. (1908). Mathematical logic as based on the theory of types. American Journal of Mathematics, 30(3), 222-262. · JFM 39.0085.03
[117] Sauer, T. & Majer, U. (2009). David Hilbert’s Lectures on the Foundations of Physics 1915-1927: Relativity, Quantum Theory and Epistemology. Berlin: Springer. · Zbl 1190.01027
[118] Shoenfield, J. (1954). A relative consistency proof. The Journal of Symbolic Logic, 19, 21-28. · Zbl 0055.00404
[119] 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.
[120] Simpson, S. (2009). Subsystems of Second Order Arithmetic (second edition). Cambridge: Cambridge University Press. · Zbl 1181.03001
[121] Smorynski, C. (1977). The incompleteness theorems. In Barwise, J., editor. Handbook of Mathematical Logic. Amsterdam: North-Holland, pp. 821-865.
[122] 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. · Zbl 0554.03036
[123] Smoryński, C. (1985). Self-Reference and Modal Logic. Amsterdam: Springer. · Zbl 0596.03001
[124] 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). · Zbl 0013.28903
[125] Tarski, A. (1956). Logic, Semantics, Metamathematics—Papers from 1923 to 1938. Oxford: Clarendon Press. · Zbl 0075.00702
[126] Tarski, A., Mostowski, A., & Robinson, R. (1953). Undecidable Theories. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland. · Zbl 0053.00401
[127] Van Heijenoort, J. (editor) (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge, MA: Harvard University Press. · Zbl 0183.00601
[128] 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. · Zbl 0915.03020
[129] Von Neumann, J. (1925). Eine axiomatisierung der mengenlehre. Journal für die reine und angewandte Mathematik, 154, 219-240. · JFM 51.0163.04
[130] Vopĕnka, P. & Hájek, P. (1972). The Theory of Semisets. Amsterdam: North-Holland. · Zbl 0332.02064
[131] 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. · Zbl 0156.25003
[132] 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.
[133] Wang, H. (1955). Undecidable sentences generated by semantic paradoxes. Journal of Symbolic Logic, 20(1), 31-43. · Zbl 0064.24501
[134] Wang, H. (1963). A Survey of Mathematical Logic. Amsterdam: North Holland. · Zbl 0106.23603
[135] Wang, H. (1981). Popular Lectures on Mathematical Logic. Mineola: Dover. · Zbl 0847.03001
[136] Weyl, H. (1918). Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis. Leipzig: Verlag von Veit & Comp. · JFM 46.0056.11
[137] Weyl, H. (1919). Der circulus vitiosus in der heutigen Begründung der Analysis. Jahresbericht der Deutschen Mathematiker-Vereinigung, 28, 85-102. · JFM 47.0895.02
[138] Zach, R. (1999). Completeness before Post: Bernays, Hilbert, and the development of propositional logic. Bulletin of Symbolic Logic, 5(03), 331-366. · Zbl 0942.03003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.