An extended version of the first revision of this entry can be found in Zach (2006).

• Ackermann, Wilhelm, 1924, “Begründung des ‘tertium non datur’ mittels der Hilbertschen Theorie der Widerspruchsfreiheit,” Mathematische Annalen, 93: 1–36. (Scholar)
• Artemov, Sergei, 2020, “The provability of consistency,” arXiv:1902.07404. (Scholar)
• Auerbach, David, 1992, “How to say things with formalisms,” in Proof, Logic, and Formalization, Michael Detlefsen, ed., London: Routledge, 77–93. (Scholar)
• Avigad, Jeremy, 2003, “Number theory and elementary arithmetic,” Philosophia Mathematica, 11: 257–284. (Scholar)
• Bernays, Paul, 1922, “Über Hilberts Gedanken zur Grundlegung der Arithmetik,” Jahresbericht der Deutschen Mathematiker-Vereinigung, 31: 10–19. English translation in Mancosu (1998a, 215–222). (Scholar)
• –––, 1923, “Erwiderung auf die Note von Herrn Aloys Müller: Über Zahlen als Zeichen,” Mathematische Annalen, 90: 159–63. English translation in Mancosu (1998a, 223–226). (Scholar)
• –––, 1928a, “Über Nelsons Stellungnahme in der Philosophie der Mathematik,” Die Naturwissenschaften, 16: 142–45. (Scholar)
• –––, 1928b,, “Zusatz zu Hilberts Vortrag über ‘Die Grundlagen der Mathematik,’” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 6: 88–92. English translation in: van Heijenoort (1967, 485–489). (Scholar)
• –––, 1930, “Die Philosophie der Mathematik und die Hilbertsche Beweistheorie,” Blätter für deutsche Philosophie, 4: 326–67. Reprinted in Bernays (1976, 17–61). English translation in Mancosu (1998a, 234–265). (Scholar)
• –––, 1976, Abhandlungen zur Philosophie der Mathematik, Darmstadt: Wissenschaftliche Buchgesellschaft.
• Cheng, Yong, 2021, “Current Research on Gödel’s Incompleteness Theorems,” Bulletin of Symbolic Logic, 27(2): 113–67. (Scholar)
• Dean, Walter, 2015, “Arithmetical reflection and the provability of soundness,” Philosophia Mathematica, 23: 31–64, doi:10.1093/philmat/nku026 (Scholar)
• Detlefsen, Michael, 1979, “On interpreting Gödel’s second theorem,” Journal of Philosophical Logic, 8: 297–313. Reprinted with a postscript in Shanker (1988, 131–154). (Scholar)
• –––, 1986, Hilbert’s Program, Dordrecht: Reidel. (Scholar)
• –––, 1990, “On an alleged refutation of Hilbert’s program using Gödel’s first incompleteness theorem,” Journal of Philosophical Logic, 19: 343–377. (Scholar)
• –––, 2001, “What does Gödel’s second theorem say?,” Philosophia Mathematica, 9: 37–71. (Scholar)
• Eder, Günther, forthcoming, “The Bernays-Müller debate,” HOPOS: The Journal of the International Society for the History of Philosophy of Science. (Scholar)
• Ewald, William Bragg (ed.), 1996, From Kant to Hilbert. A Source Book in the Foundations of Mathematics, vol. 2, Oxford: Oxford University Press. (Scholar)
• Ewald, William Bragg and Wilfried Sieg (eds.), 2013, David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–1933, Berlin and Heidelberg: Springer. (Scholar)
• Feferman, Solomon, 1988, “Hilbert’s Program relativized: Proof-theoretical and foundational reductions,” Journal of Symbolic Logic, 53(2): 364–284. (Scholar)
• –––, 1993a, “What rests on what? The proof-theoretic analysis of mathematics,” in Philosophy of Mathematics. Proceedings of the Fifteenth International Wittgenstein-Symposium, Part 1, Johannes Czermak, ed., Vienna: Hölder-Pichler-Tempsky, 147–171. Reprinted in Feferman (1998, Ch. 10, 187–208). (Scholar)
• –––, 1993b, “Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics,” PSA 1992, 2: 442–455. Reprinted in Feferman (1998, Ch. 14, 284–298). [Preprint available online]. (Scholar)
• –––, 1998, In the Light of Logic, Oxford: Oxford University Press. (Scholar)
• –––, 2000, “Does reductive proof theory have a viable rationale?.” Erkenntnis, 53: 63–96. (Scholar)
• Franks, Curtis, 2009, The Autonomy of Mathematical Knowledge: Hilbert’s Program Revisited, Cambridge: Cambridge University Press. (Scholar)
• Ganea, Mihai, 2010, “Two (or three) notions of finitism,” Review of Symbolic Logic, 3: 119–144. (Scholar)
• Gentzen, Gerhard, 1936,, “Die Widerspruchsfreiheit der reinen Zahlentheorie,” Mathematische Annalen, 112: 493–565. English translation in Gentzen (1969, 132–213). (Scholar)
• –––, 1969, The Collected Papers of Gerhard Gentzen, Amsterdam: North-Holland. (Scholar)
• Giaquinto, Marcus, 1983, “Hilbert’s philosophy of mathematics,” British Journal for Philosophy of Science, 34: 119–132. (Scholar)
• Gödel, Kurt, 1931, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I,” Monatshefte für Mathematik und Physik, 38: 173–198. Reprinted and translated in Gödel (1986, 144–195). (Scholar)
• –––, 1958, “Über eine bisher noch nicht benütze Erweiterung des finiten standpunktes,” Dialectica, 280–287. Reprinted and translated in Gödel (1990, 217–251). (Scholar)
• –––, 1986, Collected Works, vol. 1, Oxford: Oxford University Press. (Scholar)
• –––, 1990, Collected Works, vol. 2, Oxford: Oxford University Press. (Scholar)
• –––, 2003, Collected Works, vol. 4, Oxford: Oxford University Press. (Scholar)
• Hallett, Michael, 1990, “Physicalism, reductionism and Hilbert,” in Physicalism in Mathematics, Andrew D. Irvine, ed., Dordrecht: Reidel, 183–257. (Scholar)
• Hilbert, David, 1899, “Grundlagen der Geometrie,” in Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen, Leipzig: Teubner, 1–92, 1st ed. (Scholar)
• –––, 1900a, “Mathematische Probleme,” Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys. Klasse, 253–297. Lecture given at the International Congress of Mathematicians, Paris, 1900. Partial English translation in Ewald (1996, 1096–1105). (Scholar)
• –––, 1900b, “Über den Zahlbegriff,” Jahresbericht der Deutschen Mathematiker-Vereinigung, 8: 180-184. English translation in Ewald (1996, 1089–1096). (Scholar)
• –––, 1905, “Über die Grundlagen der Logik und der Arithmetik,” in Verhandlungen des dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904, A. Krazer, ed., Leipzig: Teubner, 174–85. English translation in van Heijenoort (1967, 129–138). (Scholar)
• –––, 1918a, “Axiomatisches Denken,” Mathematische Annalen, 78: 405–15. Lecture given at the Swiss Society of Mathematicians, 11 September 1917. Reprinted in Hilbert (1935, 146–156). English translation in Ewald (1996, 1105–1115). (Scholar)
• –––, 1918b, “Prinzipien der Mathematik,” Lecture notes by Paul Bernays. Winter-Semester 1917/18. Typescript. Bibliothek, Mathematisches Institut, Universität Göttingen. Edited in Ewald and Sieg (2013, 59–221).. (Scholar)
• –––, 1922a, “Grundlagen der Mathematik,” Vorlesung, Winter-Semester 1921/22. Lecture notes by Paul Bernays. Typescript. Bibliothek, Mathematisches Institut, Universität Göttingen. Edited in Ewald and Sieg (2013, 431–527). (Scholar)
• –––, 1922b, “Neubegründung der Mathematik: Erste Mitteilung,” Abhandlungen aus dem Seminar der Hamburgischen Universität, 1: 157–177. Series of talks given at the University of Hamburg, July 25–27, 1921. Reprinted with notes by Bernays in Hilbert (1935, 157–177). English translation in Mancosu (1998a, 198–214) and Ewald (1996, 1115–1134). (Scholar)
• –––, 1923, “Die logischen Grundlagen der Mathematik,” Mathematische Annalen, 88: 151–165. Lecture given at the Deutsche Naturforscher-Gesellschaft, September 1922. Reprinted in Hilbert (1935, 178–191). English translation in Ewald (1996, 1134–1148). (Scholar)
• –––, 1926, “Über das Unendliche,” Mathematische Annalen, 95: 161–190. Lecture given Münster, 4 June 1925. English translation in van Heijenoort (1967, 367–392). (Scholar)
• –––, 1928, “Die Grundlagen der Mathematik,” Abhandlungen aus dem Seminar der Hamburgischen Universität, 6: 65–85. Reprinted in Ewald and Sieg (2013, 917–942). English translation in van Heijenoort (1967, 464–479). (Scholar)
• –––, 1929, “Probleme der Grundlegung der Mathematik,” Mathematische Annalen, 102: 1–9. Lecture given at the International Congress of Mathematicians, 3 September 1928. Reprinted in Ewald and Sieg (2013, 954–966). English translation in Mancosu (1998a, 227–233). (Scholar)
• –––, 1931a, “Beweis des Tertium non datur,” Nachrichten der Gesellschaft der Wissenschaften zu Göttingen. Math.-phys. Klasse, 120-125. Reprinted in Ewald and Sieg (2013, 967–982). (Scholar)
• –––, 1931b, “Die Grundlegung der elementaren Zahlenlehre,” Mathematische Annalen, 104: 485–494. Reprinted in Hilbert (1935, 192–195) and Ewald and Sieg (2013, 983–990). English translation in Ewald (1996, 1148–1157). (Scholar)
• –––, 1935, Gesammelte Abhandlungen, vol. 3, Berlin: Springer. (Scholar)
• –––, 1992, Natur und mathematisches Erkennen, Basel: Birkhäuser. Vorlesungen, 1919–20. (Scholar)
• Hilbert, David and Ackermann, Wilhelm, 1928, Grundzüge der theoretischen Logik, Berlin: Springer. (Scholar)
• Hilbert, David and Bernays, Paul, 1923, “Logische Grundlagen der Mathematik,” Vorlesung, Winter-Semester 1922-23. Lecture notes by Paul Bernays, with handwritten notes by Hilbert. Hilbert-Nachlaß, Niedersächsische Staats- und Universitätsbibliothek, Cod. Ms. Hilbert 567. (Scholar)
• –––, 1934, Grundlagen der Mathematik, vol. 1, Berlin: Springer. (Scholar)
• –––, 1939, Grundlagen der Mathematik, vol. 2, Berlin: Springer. (Scholar)
• Hofweber, Thomas, 2000, “Proof-theoretic reduction as a philosopher’s tool,” Erkenntnis, 53: 127–146. (Scholar)
• Ignjatovic, Aleksandar, 1994, “Hilbert’s program and the omega rule,” Journal of Symbolic Logic, 59: 322–343. (Scholar)
• Incurvati, Luca, 2019, “On the concept of finitism,” Synthese, 192: 2413–2436. (Scholar)
• Kish-Bar-On, Kati, 2021, “Towards a New Philosophical Perspective on Hermann Weyl’s Turn to Intuitionism,” Science in Context, 34: 51–68. (Scholar)
• Kitcher, Philip, 1976, “Hilbert’s epistemology,” Philosophy of Science, 43: 99–115. (Scholar)
• Kreisel, Georg, 1960, “Ordinal logics and the characterization of informal notions of proof,” in Proceedings of the International Congress of Mathematicians. Edinburgh, 14–21 August 1958, J. A. Todd, ed., Cambridge: Cambridge University Press, 289–299. (Scholar)
• –––, 1968, “A survey of proof theory,” Journal of Symbolic Logic, 33: 321–388.
• –––, 1970, “Principles of proof and ordinals implicit in given concepts,” in Intuitionism and Proof Theory, A. Kino, J. Myhill, and R. E. Veseley, eds., Amsterdam: North-Holland. (Scholar)
• –––, 1983, “Hilbert’s programme,” in Philosophy of Mathematics, Paul Benacerraf and Hilary Putnam, eds., Cambridge: Cambridge University Press, 207–238, 2nd ed. (Scholar)
• Kripke, Saul A., forthcoming, “The Collapse of the Hilbert Program: A Variation on the Gödelian Theme,” Bulletin of Symbolic Logic. (Scholar)
• Mancosu, Paolo (ed.), 1998a, From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, Oxford: Oxford University Press. (Scholar)
• Mancosu, Paolo, 1998b, “Hilbert and Bernays on Metamathematics,” in (Mancosu, 1998a), 149–188. Reprinted in Mancosu (2010). (Scholar)
• –––, 1999, “Between Russell and Hilbert: Behmann on the foundations of mathematics,” Bulletin of Symbolic Logic, 5(3): 303–330. Reprinted in Mancosu (2010). (Scholar)
• –––, 2010, The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900–1940, Oxford: Oxford University Press. (Scholar)
• Mancosu, Paolo, Sergio Galvan, and Richard Zach, 2021, An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs. Oxford: Oxford University Press. (Scholar)
• Mancosu, Paolo and Ryckman, Thomas, 2002, “Mathematics and phenomenology: The correspondence between O. Becker and H. Weyl,” Philosophia Mathematica, 10: 130–202. Reprinted in Mancosu (2010). (Scholar)
• McCarthy, T., 2016, “Gödel’s third incompleteness theorem,” Dialectica 70: 87–112.
• Parsons, Charles, 1998, “Finitism and intuitive knowledge,” in The Philosophy of Mathematics Today, Matthias Schirn, ed., Oxford: Oxford University Press, 249–270. (Scholar)
• –––, 2007, Mathematical Thought and its Objects, Cambridge: Cambridge University Press. (Scholar)
• Patton, Lydia, 2014, “Hilbert’s objectivity,” Historia Mathematica, 41(2): 188–203. (Scholar)
• Peckhaus, Volker, 1990, Hilbertprogramm und Kritische Philosophie, Göttingen: Vandenhoeck und Ruprecht. (Scholar)
• Poincaré, Henri, 1906, “Les mathématiques et la logique,” Revue de métaphysique et de morale, 14: 294–317. English translation in Ewald (1996, 1038–1052). (Scholar)
• Resnik, Michael D., 1974, “On the philosophical significance of consistency proofs,” Journal of Philosophical Logic, 3: 133–47. (Scholar)
• –––, 1980, Frege and the Philosophy of Mathematics, Ithaca: Cornell University Press.
• Santos, Paulo Guilherme, Wilfried Sieg, and Reinhard Kahle, forthcoming, “A New Perspective on Completeness and Finitist Consistency,” Journal of Logic and Computation. (Scholar)
• Schirn, Matthias, 2019, “The finite and the infinite: On Hilbert’s formalist approach before and after Gödel’s incompleteness theorems,” Logique et Analyse, 245: 1–34. (Scholar)
• Schirn, Matthias, and Karl-Georg Niebergall, 2001, “Extensions of the finitist point of view,” History and Philosophy of Logic, 22(3): 135–61. (Scholar)
• Shanker, Stuart G., 1988, Gödel’s Theorem in Focus, London: Routledge.
• Sieg, Wilfried, 1990, “Reflections on Hilbert’s program,” in Acting and Reflecting, Wilfried Sieg, ed., Dordrecht: Kluwer, 171–82. Reprinted in Sieg (2013). (Scholar)
• –––, 1999, “Hilbert’s programs: 1917–1922,” Bulletin of Symbolic Logic, 5(1): 1–44. Reprinted in Sieg (2013). (Scholar)
• –––, 2013, Hilbert’s Programs and Beyond, New York: Oxford University Press. (Scholar)
• Simpson, Stephen G., 1988, “Partial realizations of Hilbert’s program,” Journal of Symbolic Logic, 53(2): 349–363. (Scholar)
• –––, 1999, Subsystems of Second Order Arithmetic, Berlin: Springer. (Scholar)
• Smorynski, Craig, 1977, “The incompleteness theorems,” in Handbook of Mathematical Logic, Jon Barwise, ed., Amsterdam: North-Holland, 821–865. (Scholar)
• Steiner, Mark, 1975, Mathematical Knowledge, Ithaca: Cornell University Press. (Scholar)
• –––, 1991, “Review of Hilbert’s Program: An Essay on Mathematical Instrumentalism (Detlefsen, 1986),” Journal of Philosophy, 88(6): 331–336. (Scholar)
• Tait, W. W., 1981, “Finitism,” Journal of Philosophy, 78: 524–546. Reprinted in Tait (2005a, 21–42). (Scholar)
• –––, 2002, “Remarks on finitism,” in Reflections on the Foundations of Mathematics. Essays in Honor of Solomon Feferman, Wilfried Sieg, Richard Sommer, and Carolyn Talcott, eds., Association for Symbolic Logic, LNL 15. Reprinted in Tait (2005a, 43–53). [Preprint available online] (Scholar)
• –––, 2005a, The Provenance of Pure Reason: Essays in the Philosophy of Mathematics and its History, New York: Oxford University Press. (Scholar)
• –––, 2005b, “Appendix to Chapters 1 and 2,” in Tait (2005a, 54–60). (Scholar)
• –––, 2019, “What Hilbert and Bernays meant by ‘finitism,’” in Philosophy of Logic and Mathematics: Proceedings of the 41st International Ludwig Wittgenstein Symposium, Gabriele M. Mras, Paul Weingartner, and Bernhard Ritter (eds.), Berlin: De Gruyter, 249–62. (Scholar)
• Takeuti, Gaisi, 1987, Proof Theory (Studies in Logic: 81), Amsterdam: North-Holland, 2nd edition. (Scholar)
• Thomas-Bolduc, Aaron, and Eamon Darnell, 2022, “Takeuti’s well-ordering proof: An accessible recontruction,” The Australasian Journal of Logic, 19(1): 1–31. (Scholar)
• van Heijenoort, Jean (ed.), 1967, From Frege to Gödel. A Source Book in Mathematical Logic, 1897–1931, Cambridge, Mass.: Harvard University Press. (Scholar)
• von Neumann, Johann, 1927, “Zur Hilbertschen Beweistheorie,” Mathematische Zeitschrift, 26: 1–46. (Scholar)
• Weyl, Hermann, 1921, “Über die neue Grundlagenkrise der Mathematik,” Mathematische Zeitschrift, 10: 37–79. Reprinted in Weyl (1968, 143–180). English translation in Mancosu (1998a, 86–118). (Scholar)
• –––, 1925, “Die heutige Erkenntnislage in der Mathematik,” Symposion, 1: 1–23. Reprinted in: Weyl (1968, 511–42). English translation in: Mancosu (1998a, 123–42). (Scholar)
• –––, 1928, “Diskussionsbemerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 6: 86–88. English translation in van Heijenoort (1967, 480–484). (Scholar)
• –––, 1968, Gesammelte Abhandlungen, vol. 1, Berlin: Springer Verlag. (Scholar)
• Zach, Richard, 1999, “Completeness before Post: Bernays, Hilbert, and the development of propositional logic,” Bulletin of Symbolic Logic, 5(3): 331–366. (Scholar)
• –––, 2003, “The practice of finitism. Epsilon calculus and consistency proofs in Hilbert’s Program,” Synthese, 137: 211–259. (Scholar)
• –––, 2004, “Hilbert’s ‘Verunglückter Beweis,’ the first epsilon theorem, and consistency proofs,” History and Philosophy of Logic, 25: 79–94. (Scholar)
• –––, 2006, “Hilbert’s program then and now,” in: Dale Jacquette, ed., Philosophy of Logic. Handbook of the Philosophy of Science, vol. 5. Amsterdam: Elsevier, 411–447. (Scholar)