Aczel, Peter. An introduction to inductive definitions. In Barwise, Jon, editor, The Handbook of Mathematical Logic, pages 739–782. North-Holland Publishing Co., Amsterdam, 1977.CrossRefGoogle Scholar

Andrews, Peter B. An Introduction to Mathematical Logic and Type Theory: To Truth through Proof. Kluwer Academic Publishers, Dordrecht, second edition, 2002.Google Scholar

Aschieri, Federico. On natural deduction for Herbrand constructive logics III: The strange case of the intuitionistic logic of constant domains. In Berardi, Stefano and Miquel, Alexandre, editors, Classical Logic and Computation, pages 1–9. Electronic Proceedings in Theoretical Compututer Science (EPTCS), Oxford, UK, 2018.Google Scholar

Avigad, Jeremy. Forcing in proof theory. The Bulletin of Symbolic Logic, 10(3):305–333, 2004.CrossRefGoogle Scholar

Avigad, Jeremy. Foundations. In Blanchette, Jasmin and Mahboubi, Assia, editors, Proof Assistants and Their Use in Mathematics and Computer Science. Springer-Verlag, in press.Google Scholar

Avigad, Jeremy and Brattka, Vasco. Computability and analysis: The legacy of Alan Turing. In Downey, Rod, editor, Turing’s Legacy: Developments from Turing’s Ideas in Logic, pages 1–47. Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2014.Google Scholar

Avigad, Jeremy and Feferman, Solomon. Gödel’s functional (“Dialectica”) interpretation. In Buss, Samuel R., editor, The Handbook of Proof Theory, pages 337–405. North-Holland Publishing Co., Amsterdam, 1998.Google Scholar

Avigad, Jeremy and Zach, Richard. The epsilon calculus. In Zalta, Edward N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, 2002.Google Scholar

Awodey, Steve. Category Theory. Oxford University Press, Oxford, second edition, 2010.Google Scholar

Baader, Franz and Nipkow, Tobias. Term Rewriting and All That. Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar

Baaz, Matthias, Egly, Uwe, and Leitsch, Alexander. Normal form transformations. In Robinson, John Alan and Voronkov, Andrei, editors, Handbook of Automated Reasoning, pages 273–333. Elsevier and MIT Press, Amsterdam, 2001.CrossRefGoogle Scholar

Baaz, Matthias, Hetzl, Stefan, and Weller, Daniel. On the complexity of proof deskolemization. The Journal of Symbolic Logic, 77(2):669–686, 2012.Google Scholar

Bachmair, Leo and Ganzinger, Harald. Resolution theorem proving. In Robinson, John Alan and Voronkov, Andrei, editors, Handbook of Automated Reasoning, pages 19–99. Elsevier and MIT Press, Amsterdam, 2001.Google Scholar

Balbes, Raymond and Dwinger, Philip. Distributive Lattices. University of Missouri Press, Columbia, MO, 1974.Google Scholar

Barendregt, Henk. The Lambda Calculus: Its Syntax and Semantics. North-Holland Publishing Co., Amsterdam and New York, 1981.Google Scholar

Barendregt, Henk, Dekkers, Wil, and Statman, Richard. Lambda Calculus with Types. Association for Symbolic Logic, Ithaca, NY; Cambridge University Press, Cambridge, 2013.Google Scholar

Barwise, Jon, editor. The Handbook of Mathematical Logic. North-Holland Publishing Co., Amsterdam, 1977.Google Scholar

Basu, Saugata, Pollack, Richard, and Roy, Marie-Françoise. Algorithms in Real Algebraic Geometry. Springer-Verlag, Berlin, second edition, 2006.Google Scholar

Beeson, Michael J. Foundations of Constructive Mathematics. Springer-Verlag, Berlin, 1985.Google Scholar

Bell, John Lane and Slomson, Alan B. Models and Ultraproducts: An Introduction. North-Holland Publishing Co., Amsterdam and London, 1969.Google Scholar

Bergman, George M. An Invitation to General Algebra and Universal Constructions. Springer-Verlag, Cham, second edition, 2015.Google Scholar

Biere, Armin, Heule, Marijn, van Maaren, Hans, and Walsh, Toby, editors. Handbook of Satisfiability. IOS Press, Amsterdam, second edition, 2021.Google Scholar

Bishop, Errett and Bridges, Douglas. Constructive Analysis. Springer-Verlag, Berlin, 1985.CrossRefGoogle Scholar

Blanchette, Jasmin and Mahboubi, Assia, editors. Proof Assistants and Their Use in Mathematics and Computer Science. Springer-Verlag, in press.Google Scholar

Blekherman, Grigoriy, Parrilo, Pablo A., and Thomas, Rekha R., editors. Semidefinite Optimization and Convex Algebraic Geometry. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013.Google Scholar

Bochnak, Jacek, Coste, Michel, and Roy, Marie-Françoise. Real Algebraic Geometry. Springer-Verlag, Berlin, 1998.Google Scholar

Boolos, George. Logic, Logic, and Logic. Harvard University Press, Cambridge, MA, 1998.Google Scholar

Börger, Egon, Grädel, Erich, and Gurevich, Yuri. The Classical Decision Problem. Springer-Verlag, Berlin, 2001.Google Scholar

Boyer, Carl B. The History of the Calculus and Its Conceptual Development. Dover Publications, Inc., New York, 1959.Google Scholar

Bradley, Aaron R. and Manna, Zohar. The Calculus of Computation: Decision Procedures with Applications to Verification. Springer-Verlag, Berlin, 2007.Google Scholar

Burr, Wolfgang. Fragments of Heyting arithmetic. The Journal of Symbolic Logic, 65(3):1223–1240, 2000.Google Scholar

Buss, Samuel R., editor. The Handbook of Proof Theory. North-Holland Publishing Co., Amsterdam, 1998a.Google Scholar

Buss, Samuel R. First-order proof theory of arithmetic. In Buss, Samuel R., editor, The Handbook of Proof Theory, pages 79–147. North-Holland Publishing Co., Amsterdam, 1998b.Google Scholar

Buss, Samuel R. An introduction to proof theory. In Buss, Samuel R., editor, The Handbook of Proof Theory, pages 1–78. North-Holland Publishing Co., Amsterdam, 1998c.Google Scholar

Butz, Carsten and Johnstone, Peter. Classifying toposes for first-order theories. Annals of Pure and Applied Logic, 91(1):33–58, 1998.Google Scholar

Carneiro, Mario. The Type Theory of Lean. Master’s thesis, Carnegie Mellon University, 2019.Google Scholar

Chang, Chen Chung and Keisler, H. Jerome. Model Theory. North-Holland Publishing Co., Amsterdam, third edition, 1990.Google Scholar

Charguéraud, Arthur. The locally nameless representation. Journal of Automated Reasoning, 49(3): 363–408, 2012.Google Scholar

Cook, Stephen and Urquhart, Alasdair. Functional interpretations of feasibly constructive arithmetic. Annals of Pure and Applied Logic, 63(2):103–200, 1993.CrossRefGoogle Scholar

Cooper, S. Barry. Computability Theory. Chapman & Hall/CRC Press, Boca Raton, FL, 2004.Google Scholar

Coquand, Thierry and Hofmann, Martin. A new method for establishing conservativity of classical systems over their intuitionistic version. Mathematical Structures in Computer Science, 9(4):323–333, 1999.Google Scholar

Cutland, Nigel. Computability: An Introduction to Recursive Function Theory. Cambridge University Press, Cambridge and New York, 1980.Google Scholar

Davey, Brian A. and Priestley, Hilary A. Introduction to Lattices and Order. Cambridge University Press, New York, second edition, 2002.Google Scholar

Davis, Martin. Computability and Unsolvability. McGraw-Hill, New York, Toronto, and London, 1958.Google Scholar

Davis, Martin. The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Dover Publications, Inc. New york, 2004.Google Scholar

Devlin, Keith. The Joy of Sets: Fundamentals of Contemporary Set Theory. Springer-Verlag, New York, second edition, 1993.Google Scholar

Diamondstone, David E., Dzhafarov, Damir D., and Soare, Robert I. П⁰ classes, Peano arithmetic, randomness, and computable domination. Notre Dame Journal of Formal Logic, 51(1):127–159, 2010.Google Scholar

Diener, Hannes. Constructive Reverse Mathematics. Habilitation, Universität Siegen, 2018.Google Scholar

Diller, Justus and Nahm, Werner. Eine Variante zur Dialectica-Interpretation der Heyting-Arithmetik endlicher Typen. Archiv für Mathematische Logik und Grundlagenforschung, 16(1):49–66, 1974.Google Scholar

Downey, Rodney G. and Hirschfeldt, Denis R. Algorithmic Randomness and Complexity. Springer-Verlag, New York, 2010.Google Scholar

Dyckhoff, Roy. Contraction-free sequent calculi for intuitionistic logic. The Journal of Symbolic Logic, 57(3):795–807, 1992.Google Scholar

Dyckhoff, Roy. Contraction-free sequent calculi for intuitionistic logic: A correction. The Journal of Symbolic Logic, 83(4):1680–1682, 2018.Google Scholar

Ebbinghaus, Heinz-Dieter and Flum, Jörg. Finite Model Theory. Springer-Verlag, Berlin, enlarged edition, 2006.Google Scholar

Enderton, Herbert B. Elements of Set Theory. Academic Press [Harcourt Brace Jovanovich, Publishers], New York and London, 1977.Google Scholar

Escardó, Martín and OlivaPaulo, . Selection functions, bar recursion and backward induction. Mathematical Structures in Computer Science, 20(2):127–168, 2010.CrossRefGoogle Scholar

Escardó, Martín and Oliva, Paulo. The Herbrand functional interpretation of the double negation shift. The Journal of Symbolic Logic, 82(2):590–607, 2017.Google Scholar

Ewald, William. From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Clarendon Press, Oxford, 1996.Google Scholar

Feferman, Solomon. Theories of finite type related to mathematical practice. In Barwise, Jon, editor, The Handbook of Mathematical Logic, pages 913–971. North-Holland Publishing Co., Amsterdam, 1977.Google Scholar

Feferman, Solomon. Definedness. Erkenntnis, 43(3):295–320, 1995.CrossRefGoogle Scholar

Feferman, Solomon. Harmonious logic: Craig’s interpolation theorem and its descendants. Synthese, 164(3):341–357, 2008.Google Scholar

Feferman, Solomon, Parsons, Charles, and Simpson, Steven G., editors. Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2010.Google Scholar

Ferreira, Gilda and Oliva, Paulo. On the relation between various negative translations. In Berger, Ulrich and Schwichtenberg, Helmut, editors, Logic, Construction, Computation, pages 227–258. Ontos Verlag, Heusenstamm, 2012.Google Scholar

Fitting, Melvin Chris. Intuitionistic Logic, Model Theory and Forcing. North-Holland Publishing Co., Amsterdam and London, 1969.Google Scholar

Franzén, Torkel. Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse. A K Peters, Ltd., Wellesley, MA, 2005.CrossRefGoogle Scholar

Friedman, Harvey. The disjunction property implies the numerical existence property. Proceedings of the National Academy of Sciences, 72(8):2877–2878, 1975.Google Scholar

Fujiwara, Makoto and Kohlenbach, Ulrich. Interrelation between weak fragments of double negation shift and related principles. The Journal of Symbolic Logic, 83(3):991–1012, 2018.Google Scholar

Gallier, Jean H. On Girard’s “candidats de reducibilité.” Technical report, University of Pennsylvania, 1989.Google Scholar

Gentzen, Gerhard. The Collected Papers of Gerhard Gentzen. North-Holland Publishing Co., Amsterdam and London, 1969. Edited by Szabo, M. E..Google Scholar

Girard, Jean-Yves, Taylor, Paul, and Lafont, Yves. Proofs and Types. Cambridge University Press, Cambridge, 1989.Google Scholar

Gödel, Kurt. Collected Works. Oxford University Press, New York, 1986–2003. Feferman, Solomon et al. eds. Volumes I–V.Google Scholar

Goodstein, Reuben L. Recursive Number Theory: A Development of Recursive Arithmetic in a Logic-Free Equation Calculus. North-Holland Publishing Co., Amsterdam, 1957.Google Scholar

Goodstein, Reuben L. Recursive Analysis. North-Holland Publishing Co., Amsterdam, 1961.Google Scholar

Görnemann, Sabine. A logic stronger than intuitionism. The Journal of Symbolic Logic, 36(2):249–261, 1971.Google Scholar

Griffor, Edward and Rathjen, Michael. The strength of some Martin–Löf type theories. Archive for Mathematical Logic, 33(5):347–385, 1994.Google Scholar

Griffor, Edward R., editor. Handbook of Computability Theory. North-Holland Publishing Co., Amsterdam, 1999.Google Scholar

Hähnle, Reiner. Tableaux and related methods. In Robinson, John Alan and Voronkov, Andrei, editors, Handbook of Automated Reasoning, pages 100–178. Elsevier and MIT Press, Amsterdam, 2001.Google Scholar

Hájek, Petr and Pudlák, Pavel. Metamathematics of First-Order Arithmetic. Springer-Verlag, Berlin, 1998. Reprint.Google Scholar

Harper, Robert. Practical Foundations for Programming Languages. Cambridge University Press, Cambridge, second edition, 2016.Google Scholar

Harrison, John. Handbook of Practical Logic and Automated Reasoning. Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar

Hindley, J. Roger and Seldin, Jonathan P. Lambda-Calculus and Combinators: An Introduction. Cambridge University Press, Cambridge, 2008.CrossRefGoogle Scholar

Hirschfeldt, Denis R. Slicing the Truth: On the Computable and Reverse Mathematics of Combinatorial Principles. World Scientific, Hackensack, NJ, 2015.Google Scholar

Hodges, Wilfrid. Model Theory. Cambridge University Press, Cambridge, 1993.Google Scholar

Ishihara, Hajime. Constructive reverse mathematics: Compactness properties. In Crosilla, Laura and Schuster, Peter, editors, From Sets and Types to Topology and Analysis, pages 245–267. Oxford University Press, Oxford, 2005.Google Scholar

Jech, Thomas. Set theory. Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded.Google Scholar

Jech, Thomas J. The Axiom of Choice. North-Holland Publishing Co., Amsterdam and London, 1973.Google Scholar

Joachimski, Felix and Matthes, Ralph. Short proofs of normalization for the simply typed λ-calculus, permutative conversions and Gödel’s T. Archive for Mathematical Logic, 42(1):59–87, 2003.Google Scholar

Johnstone, Peter T. Stone Spaces. Cambridge University Press, Cambridge, 1986. Reprint of the 1982 edition.Google Scholar

Kanamori, Akihiro. The Higher Infinite. Springer-Verlag, Berlin, second edition, 2003.Google Scholar

Kaye, Richard. Models of Peano Arithmetic. The Clarendon Press, Oxford University Press, New York, 1991.Google Scholar

Kechris, Alexander S. Classical Descriptive Set Theory. Springer-Verlag, New York, 1995.Google Scholar

Kleene, Stephen Cole. Introduction to Metamathematics. D. Van Nostrand Co., New York, 1952.Google Scholar

Kleene, Stephen Cole. Realizability: A retrospective survey. In Mathias, A. R. D and Rogers, H., editors, Cambridge Summer School in Mathematical Logic, pages 95–112. Springer-Verlag, Berlin, 1973.Google Scholar

Kneale, William and Kneale, Martha. The Development of Logic. Clarendon Press, Oxford, 1962.Google Scholar

Kohlenbach, Ulrich. Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer-Verlag, Berlin, 2008.Google Scholar

Krajícˇek, Jan. Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge University Press, Cambridge, 1995.Google Scholar

Kreisel, Georg and Krivine, Jean-Louis. Elements of Mathematical Logic. Model Theory. North-Holland Publishing Co., Amsterdam, 1967.Google Scholar

Kroening, Daniel and Strichman, Ofer. Decision Procedures: An Algorithmic Point of View. Springer-Verlag, Berlin, second edition, 2016.CrossRefGoogle Scholar

Kunen, Kenneth. Set Theory. College Publications, London, 2011.Google Scholar

Lambek, Joachim and Scott, Philip J. Introduction to Higher-Order Categorical Logic. Cambridge University Press, Cambridge, 1988.Google Scholar

Libkin, Leonid. Elements of Finite Model Theory. Springer-Verlag, Berlin, 2004.Google Scholar

Mac Lane, Saunders and Moerdijk, Ieke. Sheaves in Geometry and Logic. Springer-Verlag, New York, 1992.Google Scholar

Mancosu, Paolo. The Philosophy of Mathematical Practice. Oxford University Press, Oxford, 2008.Google Scholar

Mancosu, Paolo, Zach, Richard, and Badesa, Calixto. The development of mathematical logic from Russell to Tarski, 1900–1935. In Haaparanta, Leila, editor, The Development of Modern Logic, pages 318–470. Oxford University Press, Oxford, 2009.Google Scholar

Mancosu, Paolo, Galvan, Sergio, and Zach, Richard. An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs. Oxford University Press, Oxford, 2021.Google Scholar

Mansfield, Richard and Weitkamp, Galen. Recursive Aspects of Descriptive Set Theory. The Clarendon Press, Oxford University Press, New York, 1985.Google Scholar

Marker, David. Model Theory: An Introduction. Springer-Verlag, New York, 2002.Google Scholar

Martin-Löf, Per. Intuitionistic Type Theory. Bibliopolis, Naples, 1984.Google Scholar

Mathias, Adrian R. D. The strength of Mac Lane set theory. Annals of Pure and Applied Logic, 110(1–3):107–234, 2001.Google Scholar

Matiyasevich, Yuri V. Hilbert’s Tenth Problem. MIT Press, Cambridge, MA, 1993. Translated from the 1993 Russian original by the author.Google Scholar

McCune, William, Veroff, Robert, Fitelson, , Branden, et al. Short single axioms for Boolean algebra. Journal of Automated Reasoning, 29(1):1–16, 2002.Google Scholar

Mendelson, Elliott. Introduction to Mathematical Logic. CRC Press, Boca Raton, FL, sixth edition, 2015.Google Scholar

Mines, Ray, Richman, Fred, and Ruitenburg, Wim. A Course in Constructive Algebra. Springer-Verlag, New York, 1988.Google Scholar

Mints, Grigori. Axiomatization of a Skolem function in intuitionistic logic. In er, Martina, Kaufmann, Stefan, and Pauly, Marc, editors, Formalizing the Dynamics of Information, pages 105–114. Center for the Study of Language and Information, Stanford, 2000a.Google Scholar

Mints, Grigori. A Short Introduction to Intuitionistic Logic. Kluwer Academic/Plenum Publishers, New York, 2000b.Google Scholar

Moschovakis, Joan. Intuitionistic logic. In Zalta, Edward N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, 2018.Google Scholar

Moschovakis, Yiannis N. Descriptive Set Theory. American Mathematical Society, Providence, RI, second edition, 2009.CrossRefGoogle Scholar

Moschovakis, Yiannis N. Kleene’s amazing second recursion theorem. The Bulletin of Symbolic Logic, 16(2):189–239, 2010.Google Scholar

Munkres, James R. Topology. Prentice Hall, Inc., Upper Saddle River, NJ, second edition, 2000.Google Scholar

Nederpelt, Rob and Geuvers, Herman. Type Theory and Formal Proof. Cambridge University Press, Cambridge, 2014.Google Scholar

Negri, Sara and von Plato, Jan. Structural Proof Theory. Cambridge University Press, Cambridge, 2001.Google Scholar

Negri, Sara and von Plato, Jan. Proof Analysis. Cambridge University Press, Cambridge, 2011.Google Scholar

Nies, André. Computability and Randomness. Oxford University Press, Oxford, 2009.Google Scholar

Nordström, Bengt, Petersson, Kent, and Smith, Jan M. Programming in Martin-Löf’s Type Theory: An Introduction. Clarendon Press, New York, 1990.Google Scholar

Odifreddi, Piergiorgio. Classical Recursion Theory. North-Holland Publishing Co., Amsterdam, 1989.Google Scholar

Odifreddi, Piergiorgio. Classical Recursion Theory. Vol. II. North-Holland Publishing Co., Amsterdam, 1999.Google Scholar

Okada, Mitsuhiro. A uniform semantic proof for cut-elimination and completeness of various first and higher order logics. Theoretical Computer Science, 281(1–2):471–498, 2002.Google Scholar

Oliva, Paulo. Unifying functional interpretations: Past and future. In Schroeder-Heister, Peter, Heinzmann, Hodges, Wilfrid, and Bour, Pierre Edouard, editors, Logic, Methodology and Philosophy of Science, pages 97–122. College Publications, London, 2014.Google Scholar

Palmgren, Erik. Constructive sheaf semantics. Mathematical Logic Quarterly, 43(3):321–327, 1997.Google Scholar

Pierce, Benjamin C. Types and Programming Languages. MIT Press, Cambridge, MA, 2002.Google Scholar

Pohlers, Wolfram. Proof Theory: The First Step into Impredicativity. Springer-Verlag, Berlin, 2009.Google Scholar

Poonen, Bjorn. Characterizing integers among rational numbers with a universal-existential formula. American Journal of Mathematics, 131(3):675–682, 2009.Google Scholar

Poonen, Bjorn. Undecidable problems: A sampler. In Kennedy, Juliette, editor, Interpreting Gödel, pages 211–241. Cambridge University Press, Cambridge, 2014.Google Scholar

Pour-El, Marian B. and Richards, J. Ian. Computability in Analysis and Physics. Springer-Verlag, Berlin, 1989.Google Scholar

Pudlák, Pavel. The lengths of proofs. In Buss, Samuel R., editor, The Handbook of Proof Theory, pages 547–637. North-Holland Publishing Co., Amsterdam, 1998.Google Scholar

Quine, Willard Van Orman. The Ways of Paradox and Other Essays. Harvard University Press, Cambridge, MA, 1997.Google Scholar

Rathjen, Michael. Constructive Zermelo-Fraenkel set theory, power set, and the calculus of constructions. In Dybjer, Peter, Lindstrom, Sten, Palmgren, Erik, and Sundholm, Göran, editors, Epistemology versus Ontology, pages 313–349. Springer, Dordrecht, 2012.Google Scholar

Robinson, John Alan and Voronkov, Andrei, editors. Handbook of Automated Reasoning. Elsevier and MIT Press, Amsterdam, 2001.Google Scholar

Robinson, Julia. Definability and decision problems in arithmetic. The Journal of Symbolic Logic, 14(2):98– 114, 1949.Google Scholar

Rose, Harvey E. Subrecursion: Functions and Hierarchies. The Clarendon Press, Oxford University Press, New York, 1984.Google Scholar

Rotman, Joseph J. An Introduction to the Theory of Groups. Springer-Verlag, New York, fourth edition, 1995.Google Scholar

Russell, Bertrand. On denoting. Mind, 14(56):479–493, 1905.Google Scholar

Schindler, Ralf. Set Theory: Exploring Independence and Truth. Springer-Verlag, Cham, 2014.Google Scholar

Schwichtenberg, Helmut. Proof theory: Some aspects of cut-elimination. In Barwise, Jon, editor, The Handbook of Mathematical Logic, pages 867–895. North-Holland Publishing Co., Amsterdam, 1977.Google Scholar

Shapiro, Stewart. Incompleteness, mechanism, and optimism. The Bulletin of Symbolic Logic, 4(3): 273–302, 1998.Google Scholar

Shapiro, Stewart, editor. The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press, Oxford, 2005.Google Scholar

Shoenfield, Joseph R. Mathematical Logic. Association for Symbolic Logic, Urbana, IL, 2001. Reprint of the 1973 second printing.Google Scholar

Sieg, Wilfried. Hilbert’s Programs and Beyond. Oxford University Press, Oxford, 2013.Google Scholar

Simpson, Stephen G. Subsystems of Second Order Arithmetic. Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY, second edition, 2009.Google Scholar

Smith, Peter. An Introduction to Gödel’s Theorems. Cambridge University Press, Cambridge, second edition, 2013.Google Scholar

Smoryn´ski, Craig. Applications of Kripke models. In Troelstra, Anne S., editor, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, pages 324–391. Springer-Verlag, Berlin and New York, 1973.Google Scholar

Smorynski, Craig. The incompleteness theorems. In Barwise, Jon, editor, The Handbook of Mathematical Logic, pages 821–865. North-Holland Publishing Co., Amsterdam, 1977.Google Scholar

Smoryn´ski, Craig. The axiomatization problem for fragments. Annals of Mathematical Logic, 14(2): 193–221, 1978.Google Scholar

Smullyan, Raymond M. Gödel’s Incompleteness Theorems. The Clarendon Press, Oxford University Press, New York, 1992.Google Scholar

Smullyan, Raymond M. First-Order Logic. Dover Publications, Inc., New York, 1995. Corrected reprint of the 1968 original.Google Scholar

Soare, Robert I. Recursively Enumerable Sets and Degrees. Springer-Verlag, Berlin, 1987.Google Scholar

Soare, Robert I. Turing Computability. Springer-Verlag, Berlin, 2016.Google Scholar

Sørensen, Morten Heine and Urzyczyn, Pavel. Lectures on the Curry–Howard Isomorphism. Elsevier, Amsterdam, 2006.Google Scholar

Takeuti, Gaisi. Proof Theory. North-Holland Publishing Co., Amsterdam, second edition, 1987.Google Scholar

Tarski, Alfred. Undecidable Theories. North-Holland Publishing Co., Amsterdam, 1968. In collaboration with Andrzej Mostowski and Raphael M. Robinson.Google Scholar

Troelstra, Anne S., editor. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer-Verlag, Berlin and New York, 1973.Google Scholar

Troelstra, Anne S. Realizability. In Buss, Samuel R., editor, The Handbook of Proof Theory, pages 407–473. North-Holland Publishing Co., Amsterdam, 1998.Google Scholar

Troelstra, Anne S. and Schwichtenberg, Helmut. Basic Proof Theory. Cambridge University Press, Cambridge, second edition, 2000.Google Scholar

Troelstra, Anne S. and van Dalen, Dirk. Constructivism in Mathematics: An Introduction. North-Holland Publishing Co., Amsterdam, 1988. Volumes 1 and 2.Google Scholar

Urquhart, Alasdair. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1(4):425–467, 1995.Google Scholar

Väänänen, Jouko. Second-order and higher-order logic. In Zalta, Edward N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, fall 2020 edition, 2020.Google Scholar

van den Dries, Lou. Alfred Tarski’s elimination theory for real closed fields. The Journal of Symbolic Logic, 53(1):7–19, 1988.Google Scholar

van Heijenoort, Jean. From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge, MA, 1967.Google Scholar

Weber, Rebecca. Computability Theory. American Mathematical Society, Providence, RI, 2012.Google Scholar

Weihrauch, Klaus. Computable Analysis: An Introduction. Springer-Verlag, Berlin, 2000.Google Scholar

Werner, Benjamin. Sets in types, types in sets. In Abdali, Martin and Ito, Takayasu, editors Theoretical Aspects of Computer Software, pages 530–546. Springer-Verlag, Berlin, 1997.Google Scholar

Wolfram, Stephen. Combinators: A Centennial View. Wolfram Media Inc., Champaign, 2021.Google Scholar

Wong, Tin Lok. Interpreting weak König’s lemma using the arithmetized completeness theorem. Proceedings of the American Mathematical Society, 144(9):4021–4024, 2016.Google Scholar

Zach, Richard. The practice of finitism: Epsilon calculus and consistency proofs in Hilbert’s program. Synthese, 137(1–2):211–259, 2003.Google Scholar