Element contents
Series: Elements in Philosophy and Logic
Gödel's Incompleteness Theorems
Published online by Cambridge University Press: 05 April 2022
Summary
This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
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- Series: Elements in Philosophy and LogicOnline ISBN: 9781108981972Publisher: Cambridge University PressPrint publication: 14 April 2022
References
People, machines and Gödel (special issue, edited by Kossak, R. ). Semiotic Studies, 34(1), 2020.Google Scholar
Norbert, A’Campo, Lizhen, Ji, and Athanase, Papadopoulos. On Grothendieck’s tame topology. In Papadopoulos, A. (ed.), Handbook of Teichmüller theory. Vol. VI, volume 27 of IRMA Lect. Math. Theor. Phys., pp. 521–533. Eur. Math. Soc., Zürich, 2016.Google Scholar
Wilhelm, Ackermann. Zum Hilbertschen Aufbau der reellen Zahlen. Math. Ann., 99(1):118–133, 1928.Google Scholar
Beklemishev, Lev D.. Gödel’s incompleteness theorems and the limits of their applicability. I. Uspekhi Mat. Nauk, 65(5(395)):61–106, 2010.Google Scholar
Garrett, Birkhoff. On the structure of abstract algebras. Math. Proc. Camb. Philos. Soc., 31(7):434–454, 1935.Google Scholar
Burgess, John P.. On the outside looking in: a caution about conservativeness. In Feferman, S., Parsons, C., and Simpson, S. G. (eds.), Kurt Gödel: essays for his centennial, vol. 33 of Lect. Notes Log., pp. 128–141. Association of Symbolic Logic, La Jolla, CA, 2010.Google Scholar
Georg, Cantor. Über eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen. J. Reine Angew. Math., 77:258–263, 1873.Google Scholar
Rudolf, Carnap. Der logische Aufbau der Welt, vol. 514 of Philosophische Bibliothek [Philosophical Library]. Felix Meiner Verlag, Hamburg, 1998. Reprint of the 1928 original and of the author’s preface to the 1961 edition.Google Scholar
Alonzo, Church. A note on the Entscheidungsproblem. J. Symb. Log., 1:40–41 (Correction 1:101–102), 1936.Google Scholar
Martin, Davis. On the theory of recursive unsolvability. Ph.D. thesis, Princeton University, 1950.Google Scholar
Martin, Davis, ed. The undecidable. Dover Publications Inc., Mineola, NY, 2004. Basic papers on undecidable propositions, unsolvable problems and computable functions. Corrected reprint of the 1965 original.Google Scholar
Martin, Davis. What did Gödel believe and when did he believe it? Bull. Symb. Log., 11(2):194–206, 2005.Google Scholar
Davis, Martin. The incompleteness theorem. Notices Amer. Math. Soc., 53(4):414–418, 2006.Google Scholar
Martin, Davis and Hilary, Putnam. A computing procedure for quantification theory. J. Assoc. Comput. Mach., 7:201–215, 1960.Google Scholar
Martin, Davis, Hilary, Putnam, and Julia, Robinson. The decision problem for exponential diophantine equations. Ann. of Math. (2), 74:425–436, 1961.Google Scholar
Richard, Dedekind. What are numbers and what should they be? RIM Monographs in Mathematics. Research Institute for Mathematics, Orono, ME, 1995. Revised, edited, and translated from the German by Pogorzelski, H., Ryan, W., and Snyder, W..Google Scholar
Michael, Detlefsen. Hilbert’s Program: an essay on mathematical instrumentalism. Springer, Boston, MA, 1986.Google Scholar
Ali, Enayat and Albert, Visser. New constructions of satisfaction classes. In Fujimoto, K., Fernández, J. M., Galinon, H, and Achourioti, T. (eds.), Unifying the philosophy of truth, vol. 36 of Log. Epistemol. Unity Sci., pp. 321–335. Springer, Dordrecht, 2015.Google Scholar
Solomon, Feferman. Arithmetization of metamathematics in a general setting. Fund. Math., 49:35–92, 1960/1961.Google Scholar
Solomon, Feferman. Transfinite recursive progressions of axiomatic theories. J. Symbol. Log., 27:259–316, 1962.Google Scholar
Solomon, Feferman. Kurt Gödel: conviction and caution. Philos. Natur., 21(2–4):546–563, 1984.Google Scholar
Solomon, Feferman. Penrose’s Gödelian argument: a review of Shadows of the Mind, by Roger Penrose. Psyche, 2(7):21–32, 1995.Google Scholar
Solomon, Feferman. In the light of logic. Logic and Computation in Philosophy. Oxford University Press, New York, 1998.Google Scholar
Solomon, Feferman. Tarski’s conceptual analysis of semantical notions. In Douglas, Patterson, ed., New essays on Tarski and philosophy, pp. 72–93. Oxford University Press, Oxford, 2008.Google Scholar
Solomon, Feferman, Harvey, M. Friedman, Penelope, Maddy, and John, R. Steel. Does mathematics need new axioms? Bull. Symb. Log., 6(4):401–446, 2000.Google Scholar
Juliet, Floyd and Aki, Kanamori. Gödel vis-à-vis Russell: logic and set theory to philosophy. In Crocco, G. and Engelen, E.-M. (eds.), Gödelian studies on the Max-Phil Notebooks, vol 1. Forthcoming.Google Scholar
Juliet, Floyd and Hilary, Putnam. A note on Wittgenstein’s “notorious paragraph” about the Gödel theorem. J. Philos., 97(11):624–632, 2000.Google Scholar
Roland, Fraïssé. Sur quelques classifications des relations, basées sur des isomorphismes restreints. II. Application aux relations d’ordre, et construction d’exemples montrant que ces classifications sont distinctes. Publ. Sci. Univ. Alger. Sér. A., 2:273–295, 1954.Google Scholar
Curtis, Franks. The autonomy of mathematical knowledge: Hilbert’s program revisited. Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Torkel, Franzén. Inexhaustibility: a non-exhaustive treatment., vol. 16 of Lect. Notes Log., Association for Symbolic Logic, Urbana, IL; A K Peters, Ltd., Wellesley, MA, 2004.Google Scholar
Harvey, M. Friedman. Higher set theory and mathematical practice. Ann. Math. Logic, 2(3):325–357, 1970/1971.Google Scholar
Harvey, M. Friedman. On the necessary use of abstract set theory. Advances in Mathematics, 41:209–280, 1981.Google Scholar
Harvey, M. Friedman. Finite functions and the necessary use of large cardinals. Ann. of Math. (2), 148(3):803–893, 1998.Google Scholar
Harvey, M. Friedman. Internal finite tree embeddings. In Sieg, W., Sommer, R., and Talcott, C. (eds.), Reflections on the foundations of mathematics, vol. 15 of Lect. Notes Log., pp. 60–91. Association for Symbolic Logic, Urbana, IL; A K Peters, Ltd., Wellesley, MA,Google Scholar
Haim, Gaifman. Naming and diagonalization, from Cantor to Gödel to Kleene. Log. J. IGPL, 14(5):709–728, 2006.Google Scholar
Robin, Gandy. The confluence of ideas in 1936. In Herken, R. (ed.), The universal Turing machine: a half-century survey, pp. 55–111. Oxford University Press, New York, 1988.Google Scholar
Gerhard, Gentzen. Die Widerspruchsfreiheit der reinen Zahlentheorie. Math. Ann., 112:493–565, 1936.Google Scholar
Kurt, Gödel. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatsh. Math. Phys., 38(1):173–198, 1931.Google Scholar
Kurt, Gödel. The consistency of the axiom of choice and of the generalized continuum hypothesis. Proc. Natl. Acad. Sci. USA, 24:556–557, 1938.Google Scholar
Kurt, Gödel. Collected works. Vol. I: Publications 1929–1936. The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by Feferman, S..Google Scholar
Kurt, Gödel. Collected works. Vol. II: Publications 1938–1974. The Clarendon Press, Oxford University Press, New York, 1990. Edited and with a preface by Feferman, S..Google Scholar
Kurt, Gödel. Remarks before the Princeton bicentennial conference of problems in mathematics, 1946. In Collected works. Vol. II: Publications 1938–1974. The Clarendon Press, Oxford University Press, New York, 1990. Edited and with a preface by Feferman, S..Google Scholar
Kurt, Gödel. Collected works. Vol. III: Unpublished essays and lectures. The Clarendon Press, Oxford University Press, New York, 1995. With a preface by Feferman, S.. Edited by Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., and Solovay, R. M..Google Scholar
Kurt, Gödel. Collected works. Vol. IV: Correspondence A–G. The Clarendon Press, Oxford University Press, Oxford, 2003. Edited by Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., and Sieg, W..Google Scholar
Kurt, Gödel. Collected works. Vol. V: Correspondence H–Z. The Clarendon Press, Oxford University Press, Oxford, 2003. Edited by Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., and Sieg, W..Google Scholar
Warren, D. Goldfarb. The Gödel class with identity is unsolvable. Bull. Amer. Math. Soc. (N.S.), 10(1):113–115, 1984.Google Scholar
Balthasar, Grabmayr. On the invariance of Gödel’s second theorem with regard to numberings. Rev. Symb. Log., 14(1):51–84, 2021.Google Scholar
Balthasar, Grabmayr and Albert, Visser. Self-reference upfront: a study of self-referential gödel numberings. Rev. Symb. Log., pp. 1–41, 2021. doi:10.1017/S1755020321000393.Google Scholar
Ronald, L. Graham, Bruce, L. Rothschild, and Joel, H. Spencer. Ramsey theory. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons, Inc., New York, 1980.Google Scholar
Ivor, Grattan-Guinness. In memoriam Kurt Gödel: his 1931 correspondence with Zermelo on his incompletability theorem. Historia Math., 6(3):294–304, 1979.Google Scholar
Robert, Gray. Georg Cantor and transcendental numbers. Amer. Math. Monthly, 101(9):819–832, 1994.Google Scholar
Fritz, Grunewald and Dan, Segal. On the integer solutions of quadratic equations. J. Reine Angew. Math., 569:13–45, 2004.Google Scholar
Petr, Hájek and Pavel, Pudlák. Metamathematics of first-order arithmetic. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1998. Second printing.Google Scholar
Volker, Halbach and Leon, Horsten. Computational structuralism. Philos. Math. (3), 13(2):174–186, 2005.Google Scholar
Volker, Halbach and Albert, Visser. Self-reference in arithmetic I. Rev. Symb. Log., 7(4):671–691, 2014.Google Scholar
Volker, Halbach and Albert, Visser. Self-reference in arithmetic II. Rev. Symb. Log., 7(4):692–712, 2014.Google Scholar
Jacques, Herbrand. Logical writings: a translation of the ıt Écrits logiques, Harvard University Press, Cambridge, MA, 1971. Edited by van Heijenoort, J. and including contributions by Chevalley, C. and Lautman, A..Google Scholar
Hilbert, David and Ackermann, Wilhelm. Grundzüge der theoretischen Logik. Die Grundlehren der mathematischen Wissenschaften Bd. 27). VIII, 120 S.J. Springer, Berlin. 1928.Google Scholar
David, Hilbert and Paul, Bernays. Grundlagen der Mathematik. I, 2nd edn. Die Grundlehren der mathematischen Wissenschaften, vol. 40. Springer-Verlag, Berlin, New York, 1968.Google Scholar
David, Hilbert and Paul, Bernays. Grundlagen der Mathematik. II, 2nd edn. Zweite Auflage. Die Grundlehren der mathematischen Wissenschaften, vol. 50. Springer-Verlag, Berlin, New York, 1970.Google Scholar
David, Hilbert. On the infinite. In van Heijenoort, J. (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, pp. 367–392. Harvard University Press, Cambridge, MA, 1965.Google Scholar
David, Hilbert. David Hilbert’s lectures on the foundations of geometry, 1891–1902, vol. 1 of David Hilbert’s Lectures on the Foundations of Mathematics and Physics 1891–1933. Springer-Verlag, Berlin, 2004. Edited by Michael, Hallett and Ulrich, Majer.Google Scholar
James, P. Jones. Three universal representations of recursively enumerable sets. J. Symb. Log., 43(2):335–351, 1978.Google Scholar
Richard, Kaye. Models of Peano arithmetic, vol. 15 of Oxford Logic Guides. The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications.Google Scholar
Keisler, H. Jerome. Ultraproducts and saturated models. Nederl. Akad. Wetensch. Proc. Ser. A 67 = Indag. Math., 26:178–186, 1964.CrossRefGoogle Scholar
Juliette, Kennedy. Turing, Gödel and the “Bright Abyss." In Floyd, J. and Bokulich, A. (eds), Philosophical Explorations of the Legacy of Alan Turing, vol. 324 of Boston Studies in Philosophy. Springer, Cham, 2017.Google Scholar
Juliette, Kennedy. Gödel’s Thesis: an appreciation. In Baaz, Mathias, Papadimitriou, Christos H., Putnam, Hilary W., Scott, Dana S., and Harper, Charles L. Jr., eds., Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press, Cambridge, 2011.Google Scholar
Juliette, Kennedy. Gödel’s 1946 Princeton Bicentennial Lecture: an appreciation. In Kennedy, Juliette, ed., Interpreting Gödel. Cambridge University Press, Cambridge, 2014.Google Scholar
Juliette, Kennedy. Kurt Gödel. In Zalta, Edward N., ed., The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Stanford, CA, winter edn. 2018.Google Scholar
Juliette, Kennedy. Gödel, Tarski and the lure of natural language: logical entanglement, formalism freeness. Cambridge University Press, Cambridge, 2020.Google Scholar
Jussi, Ketonen and Robert, Solovay. Rapidly growing Ramsey functions. Ann. of Math. (2), 113(2):267–314, 1981.Google Scholar
Stephen, C. Kleene. General recursive functions of natural numbers. Math. Ann., 112(1):727–742, 1936.Google Scholar
Stephen, C. Kleene. A symmetric form of Gödel’s theorem. Nederl. Akad. Wetensch., Proc., 53:800–802 = Indagationes Math. 12:, 244–246 (1950), 1950.Google Scholar
Peter, Koellner. On the question of absolute undecidability. In Kurt Gödel: essays for his centennial, vol. 33 of Lect. Notes Log., pp. 189–225. Association of Symbolic Logic, La Jolla, CA, 2010.Google Scholar
Henryk, Kotlarski. The incompleteness theorems after 70 years. Ann. Pure Appl. Logic, 126(1–3):125–138, 2004.Google Scholar
Georg, Kreisel. Kurt Gödel, 1906-1978. Biographical Memoirs of Fellows of the Royal Society, 26:148–224, 1980. Corrigenda, 27:697, 1981; further corrigenda, 28:697, 1982.Google Scholar
Saul, Kripke. The collapse of the Hilbert Program: why a system cannot prove its own 1-consistency. Bull. Symbolic Logic, 15(2):229–231, 2009.Google Scholar
Saul, Kripke. The Church-Turing “thesis” as a special corollary of Gödel’s completeness theorem. In Copeland, B.J., Posy, C., and Shagrir, O. (eds.), Computability—Turing, Gödel, Church, and beyond, pp. 77–104. MIT Press, Cambridge, MA, 2013.Google Scholar
Saul, Kripke. Mathematical incompleteness results in first-order peano arithmetic: a revisionist view of the early history. Hist. Philos. Logic, doi:10.1080/01445340.2021.1976052. 2021.Google Scholar
Shira, Kritchman and Ran, Raz. The surprise examination paradox and the second incompleteness theorem. Notices Amer. Math. Soc., 57(11):1454–1458, 2010.Google Scholar
Taishi, Kurahashi. A note on derivability conditions. J. Symb. Log., 85(3):1224–1253, 2020.Google Scholar
Casimir, Kuratowski. Sur l’état actuel de l’axiomatique de la théorie des ensembles. Ann. Soc. Polon. Math., 3:146–147, 1925.Google Scholar
Richard, Laver. On the consistency of Borel’s conjecture. Acta Math., 137(3-4):151–169, 1976.Google Scholar
Azriel, Lévy. Axiom schemata of strong infinity in axiomatic set theory. Pacific J. Math., 10:223–238, 1960.Google Scholar
Martin, Hugo Löb. Solution of a problem of Leon Henkin. J. Symb. Log., 20:115–118, 1955.Google Scholar
John, R. Lucas. Metamathematics and the philosophy of mind: a rejoinder. Philos. Sci., 38:310–313, 1971.Google Scholar
Nikolai, Luzin. Sur les ensembles projectifs de m. henri lebesgue. 180(2):1572–1574, 1925.Google Scholar
Angus, Macintyre. The impact of Gödel’s incompleteness theorems on mathematics. In Baaz, M., Papadimitriou, C. H., Putnam, H. W., Scott, D. S., and Harper, C. L. Jr. (eds.), Kurt Gödel and the foundations of mathematics, pp. 3–25. Cambridge University Press, Cambridge, 2011.Google Scholar
Penelope, Maddy. Defending the axioms: on the philosophical foundations of set theory. Oxford University Press, Oxford, 2011.CrossRefGoogle Scholar
Anatoly, I. Mal’tsev. Untersuchungen aus dem Gebiete der mathematischen Logik. Rec. Math. Moscou, n. Ser., 1:323–336, 1936.Google Scholar
Donald, A. Martin. Measurable cardinals and analytic games. Fund. Math., 66:287–291, 1969/1970.Google Scholar
Donald, A. Martin and John, R. Steel. Projective determinacy. Proc. Nat. Acad. Sci. U.S.A., 85(18):6582–6586, 1988.Google Scholar
Juri, V. Matijasevič. The Diophantineness of enumerable sets. Dokl. Akad. Nauk SSSR, 191:279–282, 1970.Google Scholar
Kenneth, McAloon. Consistency results about ordinal definability. Ann. Math. Logic, 2(4):449–467, 1970/1971.Google Scholar
Yiannis, N. Moschovakis. Descriptive set theory, 2nd edn., vol. 155 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2009.Google Scholar
Yiannis, N. Moschovakis. Kleene’s amazing second recursion theorem. Bull. Symbolic Logic, 16(2):189–239, 2010.Google Scholar
Andrzej, Mostowski. On recursive models of formalised arithmetic. Bull. Acad. Polon. Sci. Cl. III, 5:705–710, LXII, 1957.Google Scholar
Andrzej, Mostowski. Thirty years of foundational studies: lectures on the development of mathematical logic and the study of the foundations of mathematics in 1930–1964. Acta Philosophica Fennica, Fasc. XVII. Barnes & Noble, Inc., New York, 1966.Google Scholar
Andrzej, Mostowski. Sentences undecidable in formalized arithmetic. Greenwood Press, Westport, CT, 1982. An exposition of the theory of Kurt Gödel. Reprint of the 1952 original.Google Scholar
John, Myhill and Dana, Scott. Ordinal definability. In Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I), University of California, Los Angeles, CA, 1967), pp. 271–278. American Mathematical Society, Providence, RI, 1971.Google Scholar
Jeff, Paris and Leo, Harrington. A mathematical incompleteness in Peano arithmetic. In Handbook of mathematical logic, vol. 90 of Stud. Logic Found. Math., Barwise, Jon & Jerome Keisler, H. (eds.) pp. 1133–1142. North-Holland, Amsterdam, 1977.Google Scholar
Charles, Parsons. Platonism and mathematical intuition in Kurt Gödel’s thought. Bull. Symb. Log., 1(1):44–74, 1995.Google Scholar
Orlando, Patterson. Slavery and Social Death: A Comparative Study. Harvard University Press, Cambridge, MA, 2018.Google Scholar
Roger, Penrose. Shadows of the mind: a search for the missing science of consciousness. Oxford University Press, Oxford, 1994.Google Scholar
Panu, Raatikainen. Hilbert’s Program revisited. Synthese, 137 (special issue): 157–177, 2000.Google Scholar
Abraham, Robinson. On languages which are based on non-standard arithmetic. Nagoya Math. J., 22:83–117, 1963.Google Scholar
Julia, Robinson. Existential definability in arithmetic. Trans. Amer. Math. Soc., 72:437–449, 1952.Google Scholar
Barkley, Rosser. Extensions of some theorems of Gödel and Church. J. Symb. Log., 1(3):87–91, 1936.Google Scholar
Saeed, Salehi. On the diagonal lemma of Gödel and Carnap. Bull. Symb. Log., 26(1):80–88, 2020.Google Scholar
Saharon, Shelah. Every two elementarily equivalent models have isomorphic ultrapowers. Israel J. Math., 10:224–233, 1971.Google Scholar
Saharon, Shelah. Infinite abelian groups, Whitehead problem and some constructions. Israel J. Math., 18:243–256, 1974.Google Scholar
Waclaw, Sierpinski. Sur un ensemble non denombrable, dont toute image continue est de mesure nulle. Fund. Math., 11:302–304, 1928.Google Scholar
Stephen, G. Simpson. Nonprovability of certain combinatorial properties of finite trees. In Harrington, L., Morley, M., Sĉêdrov, A., and Simpson, S. (eds.), Harvey Friedman’s research on the foundations of mathematics, vol. 117 of Studies in Logic and the Foundations of Mathematics, pp. 87–117. North-Holland, Amsterdam, 1985.Google Scholar
Craig, Smoryński. Lectures on nonstandard models of arithmetic. In Lolli, G., Longo, G., and Marcja, A. (eds.), Logic colloquium ’82 (Florence, 1982), volume 112 of Studies in Logic and Foundations of Mathematics, pp. 1–70. North-Holland, Amsterdam, 1984.Google Scholar
Raymond, M. Smullyan. Theory of formal systems. Annals of Mathematics Studies, No. 47. Princeton University Press, Princeton, NJ, 1961.Google Scholar
William, W. Tait. Gödel on intuition and on Hilbert’s finitism. In Feferman, S., Parsons, C., and Simpson, S. G. (eds.), Kurt Gödel: essays for his centennial, vol. 33 of Lecture Notes in Logic, pp. 88–108. Association of Symbolic Logic La Jolla, CA, 2010.Google Scholar
Alfred, Tarski. Sur les ensembles définissables de nombres réels. Fund. Math., (7):210–239, 1931.Google Scholar
Alfred, Tarski. Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica, 1:261–405, 1936.Google Scholar
Alfred, Tarski. Undecidable theories. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1968. In collaboration with Andrzej Mostowski and Raphael M. Robinson, second printing.Google Scholar
Richard, Tieszen. Gödel and the intuition of concepts. Synthese, 133(3):363–391, 2002.Google Scholar
Alan, M. Turing. On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc., S2-42(1):230.Google Scholar
Alan, M. Turing. Systems of logic based on ordinals. Proc. London Math. Soc. (2), 45(3):161–228, 1939.Google Scholar
Mark van, Atten. Essays on Gödel’s reception of Leibniz, Husserl, and Brouwer, vol. 35 of Logic, Epistemology, and the Unity of Science. Springer, Cham, 2015.Google Scholar
Mark van, Atten and Juliette, Kennedy. On the philosophical development of Kurt Gödel. Bull. Symb. Log., 9(4):425–476, 2003.Google Scholar
Mark van, Atten and Juliette, Kennedy. “Gödel’s modernism: on set-theoretic incompleteness,” revisited. In Lindström, S., Palmgren, E., Segerberg, K., and Stoltenberg-Hansen, V. (eds.), Logicism, intuitionism, and formalism, vol. 341 of Synthese Library, pp. 303–355. Springer, Dordrecht, 2009.Google Scholar
Robert, L. Vaught. Alfred Tarski’s work in model theory. J. Symb. Log., 51(4):869–882, 1986.Google Scholar
Robert, L. Vaught. Errata: “Alfred Tarski’s work in model theory.” J. Symb. Log. 52(4):vii, 1987.Google Scholar
Giorgio, Venturi and Matteo, Viale. New axioms in set theory. Mat. Cult. Soc. Riv. Unione Mat. Ital. (I), 3(3):211–236, 2018.Google Scholar
Albert, Visser. From Tarski to Gödel – or how to derive the second incompleteness theorem from the undefinability of truth without self-reference. J. Logic Comput., 29(5):595–604, 2019.Google Scholar
Hao, Wang. A logical journey: representation and mind. MIT Press, Cambridge, MA, 1996.Google Scholar
Andreas, Weiermann. Phase transitions for Gödel incompleteness. Ann. Pure Appl. Logic, 157(2-3):281–296, 2009.Google Scholar
Jan, Woleński. Gödel, Tarski and the undefinability of truth. Jbuch. Kurt-Gödel-Ges., pp. 97–108 (1993), 1991.Google Scholar
Hugh, Woodin. In search of Ultimate-L: the 19th Midrasha Mathematicae Lectures. Bull. Symb. Log., 23(1):1–109, 2017.Google Scholar
Woodin, W. Hugh. Supercompact cardinals, sets of reals, and weakly homogeneous trees. Proc. Nat. Acad. Sci. U.S.A., 85(18):6587–6591, 1988.CrossRefGoogle ScholarPubMed
Richard, Zach. Hilbert’s Program. In Zalta, E. N. (ed.), The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Stanford, CA, 2003.Google Scholar
Ernst, Zermelo. Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundam. Math., 16:29–47, 1930.Google Scholar
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