Skip to main content Accessibility help
Interpreting Gödel
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 2
  • Export citation
  • Recommend to librarian
  • Buy the print book

Book description

The logician Kurt Gödel (1906–1978) published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Gödel's work on the foundations and philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues.


'These essays explore most aspects of Gödel's legacy, including his conceptions of intuition and analyticity, the Completeness theorem, the set-theoretic multiverse and the current state of mathematical logic.'

Graham Hoare Source: The Mathematical Gazette

'In sum, this is a collection of stimulating essays, mathematically as well as philosophically. They are not exactly easy reading and require familiarity, at least in broad strokes, with Gödel’s mathematical work and his central philosophical ideas (as well as their evolution and historical context). The patient reader will be rewarded by a deeper understanding of both.'

Wilfried Sieg Source: Isis

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Send to Kindle
  • Send to Dropbox
  • Send to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.



Oliver Aberth, 1971. The failure in computable analysis of a classical existence theorem for differential equations, Proc. Am. Math. Soc. 30, 151–156.
Uri Abraham and Menachem Magidor, 2010. Cardinal arithmetic. In Matthew Foreman and Akihiro Kanamori (editors), Handbook of Set Theory, volumes 1, 2, 3, pp. 1149–1227. Springer, Dordrecht.
John W. Addison, 1959. Some consequences of the axiom of constructibility, Fundam. Mathematicae 46, 337–357.
Andrew Adler, 1969. Some recursively unsolvable problems in analysis, Proc. Am. Math. Soc. 22, 523–526.
S. I. Adyan, 1957a. Unsolvability of some algorithmic problems in the theory of groups, Trudy Moskov. Mat. Obšč. 6, 231–298 (in Russian).
S. I. Adyan, 1957b. Finitely presented groups and algorithms, Dokl. Akad. Nauk SSSR (N.S.) 117, 9–12 (in Russian).
David J. Anick, 1985. Diophantine equations, Hilbert series, and undecidable spaces, Ann. Math. 122(1), 87–112.
Andrew Arana and Paolo Mancosu, 2012. On the relationship between plane and solid geometry, Rev. Symbolic Logic 5(2), 294–353.
W. W. Armstrong, 1974. Dependency structures of database relationships, Information Processing 74.
A. Arnauld and P. Nicole, 1964. The Art of Thinking, or the Port Royal Logic, translated by J. Dickoff and P. James. Bobbs-Merrill, Indianapolis, IN.
David Aspero, Paul Larson, and Justin Moore, 2013. Forcing axioms and the continuum hypothesis, Acta Mathematica 210, 1–29.
Joan Bagaria, 2006. Axioms of generic absoluteness. In Logic Colloquium 2002, volume 27 of Lecture Notes in Logic, pp. 28–47. Association of Symbolic Logic, La Jolla, CA.
John Baldwin, 2009. Categoricity, volume 50 of University Lecture Series. American Mathematical Society, Providence, RI.
John Baldwin, 2013. Formalization, primitive concepts, and purity, Rev. Symbolic Logic 6, 87–128.
St. Banach and A. Tarski, 1924. Sur la décomposition des ensembles de points en parties respectivement congruentes, Fundam. Mathematicae 6, 244–277.
Y. Bar-Hillel, editor, 1966. Essays on the Foundations of Mathematics: Dedicated to A. A. Fraenkel on his seventieth anniversary, 2nd edition. Magnum Press, Jerusalem.
Jon Barwise, editor, 1977. Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, volume 90. North-Holland, Amsterdam.
P. Benacerraf, 1973. Mathematical truth, J. Philos. 70, 661–680; also in Benacerraf and Putnam (1983), pp. 403–420.
P. Benacerraf and H. Putnam, editors, 1983. Philosophy of Mathematics: Selected Readings, 2nd edition. Cambridge University Press, Cambridge.
Robert Berger, 1966. The undecidability of the domino problem, Mem. Am. Math. Soc. 66, 72.
P. Bernays, 1918. Beiträge zur axiomatischen Behandlung des Logik-Kalküuls, Habilitationsschrift, University of Göttingen.
E. Beth, 1959. The Foundations of Mathematics; a Study in the Philosophy of Science. North-Holland, Amsterdam.
Andreas Blass and Saharon Shelah, 1987. There may be simple P1- and P2-points and the Rudin–Keisler ordering may be downward directed, Ann. Pure Appl. Logic 33, 213–243.
Vincent D. Blondel and John N. Tsitsiklis, 2000. A survey of computational complexity results in systems and control, Automatica J. IFAC 36(9), 1249–1274.
Vincent D. Blondel, Olivier Bournez, Pascal Koiran, and John N. Tsitsiklis, 2001. The stability of saturated linear dynamical systems is undecidable, J. Comput. System Sci. 62(3), 442–462.
L. M. Blumenthal, 1940. A paradox, a paradox, a most ingenious paradox, Am. Math. Monthly 47, 346–353.
B. Bolzano, 1804. Betrachtungen über einige Gegenstände der Elementargeometrie. Karl Barth, Prague; translated in part by S. Russ in Ewald (1996), pp. 172–174.
B. Bolzano, 1810. Beyträge zu einer begründeteren Darstellung der Mathematik. Caspar Widtmann, Prague; translated by S. Russ in Ewald (1996), pp. 174–224.
B. Bolzano, 1817a. Die drey Probleme der Rectification, der Complanation und der Cubirung, ohne Betrachtung des unendlich Kleinen, ohne die Annahme des Archimedes und ohne irgend eine nicht streng eweisliche Voraussetzung geloest; angleich als Probe einer ganzlichen Umstaltung der Raumwissenschaft allen Mathematikern zur Prufung vorgelegt. Gotthelf Kummer, Leipzig.
B. Bolzano, 1817b. Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwei Werten, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege. Gottlieb Haase, Prague; translated by S. Russ in Ewald (1996), pp. 225–248.
B. Bolzano, 1837. Wissenschaftslehre. Versuch einer ausführlichen und grössenteils neuen Darstellung der Logik mit steter Rücksicht auf deren bisherigen Bearbeiter. Seidel, Sulzbach; translated by B. Terrel in Bolzano (1973).
B. Bolzano, 1973. Theory of Science, edited by J. Berg. D. Reidel, Boston, MA.
G. Boole, 1854. An Investigation of the Laws of Thought. Walton and Maberly, London.
George Boolos, 1971. The iterative conception of set, J. Philos. 68(8), 215–231.
George Boolos, 1998. Must we believe in set theory? In Logic, Logic, and Logic, pp. 120–132. Harvard University Press, Cambridge, MA.
William W. Boone, 1959. The word problem, Ann. Math. 70(2), 207–265.
J. L. Borges, 1964. Other Inquisitions 1937–1952, translated by R. L. C. Simms. University of Texas Press, Austin, TX.
Robert Brandom, 2011. Platforms, patchworks, and parking garages: Wilson’s account of conceptual fine-structure in Wandering Significance, Philos. Phenomenol. Res. 82(1), 183–201.
L. E. J. Brouwer, 1913. Intuitionism and formalism, Bull. Am. Math. Soc. 20, 81–96; reprinted in Benacerraf and Putnam (1983), pp. 77–89.
Dan Brumleve, Joel David Hamkins, and Philipp Schlicht, 2012. The mate-in-n problem of infinite chess is decidable (January 31, 2012). Preprint, arXiv:1201.5597v2.
Tyler Burge, 2005. Frege on sense and linguistic meaning. In Truth, Thought, Reason, pp. 242–269. Clarendon Press, Oxford.
John P. Burgess, 2008. Mathematics, Models, and Modality. Cambridge University Press.
J. Burgess and G. Rosen, 1997. Non-empirical physics. In A Subject with No Object, pp. 118–123. Oxford University Press, Oxford.
E. Carson, 2011. Sensibility, Understanding and Number in Kant, Workshop on Knowledge, Representation and Proof in the Modern Era, University of Notre Dame, November, 2011.
E. Carson and R. Huber, editors, 2006. Intuition and the Axiomatic Method, Western Ontario Series in Philosophy of Science. Springer, Dordrecht.
Alonzo Church, 1936a. An unsolvable problem of elementary number theory, Am. J. Math. 58, 345–363.
Alonzo Church, 1936b. A note on the Entscheidungsproblem, J. Symbolic Logic 1, 40–41.
Marcus Tullius Cicero, Treatise on Topics. In The Orations of Marcus Tullius Cicero, volume IV, pp. 1894–1903, translated by C. D. Yonge. Bell and Sons, London.
Paul Cohen, 1963. The independence of the continuum hypothesis, Proc. Natl. Acad. Sci. USA 50, 1143–1148.
Paul J. Cohen, 1964. The independence of the continuum hypothesis. II, Proc. Natl. Acad. Sci. USA 51, 105–110.
Paul Cohen, 1966. Set Theory and the Continuum Hypothesis. Benjamin, New York.
Donald J. Collins, 1972. Representation of Turing reducibility by word and conjugacy problems in finitely presented groups, Acta Mathematica 128(1–2), 73–90.
A. Comte, 1892. La Philosophie Positive: Les preliminaires généraux et la philosophie mathématique, 5th edition. La Société Positiviste, Paris.
J. H. Conway, 1972. Unpredictable iterations, Proceedings of the 1972 Number Theory Conference, University of Colorado, Boulder, CO, August 14–18, 1972; reprinted in Lagarias (2010).
R. Courant and H. Robbins, 1981. What is Mathematics?Oxford University Press, Oxford.
George Csicsery, 2008. Julia Robinson and Hilbert’s tenth problem. Documentary, Zala Films,
Karel Culik II, 1996. An aperiodic set of 13 Wang tiles, Discrete Math. 160(1–3), 245–251.
Martin Davis, 1953. Arithmetical problems and recursively enumerable predicates, J. Symbolic Logic 18, 33–41.
Martin Davis, 1958. Computability and Unsolvability, McGraw-Hill Series in Information Processing and Computers. McGraw-Hill, New York.
Martin Davis, editor, 1965. The Undecidable: Basic papers on undecidable propositions, unsolvable problems and computable functions. Raven Press, Hewlett, NY.
Martin Davis, 1977. Unsolvable problems. In Jon Barwise, editor, Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, volume 90, pp. 567–594. North-Holland, Amsterdam.
Martin Davis, editor, 2004. The Undecidable: Basic papers on undecidable propositions, unsolvable problems and computable functions, corrected reprint. Dover Publications, Mineola, NY.
Martin Davis, Hilary Putnam, and Julia Robinson, 1961. The decision problem for exponential diophantine equations, Ann. Math. 2 74, 425–436.
Martin Davis, Yuri Matiyasevich, and Julia Robinson, 1976. Hilbert’s tenth problem: Diophantine equations: positive aspects of a negative solution, Mathematical Developments Arising from Hilbert Problems, pp. 323–378 (loose erratum), Proc. Symp. Pure Math., De Kalb, IL, 1974, Vol. XXVIII. American Mathematical Society, Providence, RI.
J. W. Dawson, 1993. The compactness of first-order logic: from Gödel to Lindström, History Philos. Logic 14(1), 15–37.
M. Dehn, 1911. Über unendliche diskontinuierliche Gruppen, Math. Ann. 71(1), 116–144 (in German).
J. Denef, 1980. Diophantine sets over algebraic integer rings. II, Trans. Am. Math. Soc. 257(1), 227–236.
J. Denef and L. Lipshitz, 1978. Diophantine sets over some rings of algebraic integers, J. London Math. Soc. 2 18(3), 385–391.
J. Denef and L. Lipshitz, 1984. Power series solutions of algebraic differential equations, Math. Ann. 267(2), 213–238.
J. Denef and L. Lipshitz, 1989. Decision problems for differential equations, J. Symbolic Logic 54(3), 941–950.
Jan Denef, Leonard Lipshitz, Thanases Pheidas, and Jan Van Geel, editors, 2000. Hilbert’s Tenth Problem: Relations with Arithmetic and Algebraic Geometry, Papers from the workshop held at Ghent University, Ghent, November 2–5, 1999, Contemporary Mathematics, volume 270. American Mathematical Society, Providence, RI.
Michael Detlefsen, 1979. On interpreting Gödel’s second theorem. J. Philos. Logic, 8(3), 297–313.
Michael Detlefsen, 2005. Formalism. In S. Shapiro (editor), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press, Oxford.
Diogenes Laertius, 1925. Lives of Eminent Philosophers, translated by R. D. Hicks. Loeb Classical Library, Harvard University Press, Cambridge, MA.
John E. Doner, Andrzej Mostowski, and Alfred Tarski, 1978. The elementary theory of well-ordering—a metamathematical study. In Logic Colloquium ’77 (Proc. Conf., Wroctaw, 1977), volume 96 of Studies in Logic and the Foundations of Mathematics, pp. 1–54. North-Holland, Amsterdam,.
M. Dummett, 1996. Frege and Kant on geometry. In M. Dummett (editor), Frege and Other Philosophers, pp. 126–158. Oxford University Press.
C. Duret, 1613. Trésor de l’Histoire des Langues. Cologne.
Kirsten Eisenträger, 2004. Hilbert’s tenth problem for function fields of varieties over , Int. Math. Res. Not. 59, 3191–3205.
Paul C. Eklof and Alan Mekler, 2002. Almost Free Modules: Set Theoretic Methods, volume 65 of North-Holland Mathematical Library, revised edition. North-Holland, Amsterdam.
Jörg Endrullis, Clemens Grabmayer, and Dimitri Hendriks, 2009. Complexity of Fractran and Productivity, Automated Deduction, CADE-22, Lecture Notes in Computer Science, volume 5663, pp. 371–387. Springer, Berlin.
W. Ewald, 1996. From Kant to Hilbert: A Sourcebook in the Foundations of Mathematics, volume I. Oxford University Press, New York.
S. Feferman, 1960–1961. Arithmetization of metamathematics in a general setting, Fundam. Mathematicae 49, 35–92.
Solomon Feferman, M. Harvey, M. Friedman, Penelope Maddy, and John R. Steel, 2000. Does mathematics need new axioms?Bull. Symbolic Logic 6(4), 401–446.
Solomon Feferman, Charles Parsons, and Stephen G. Simpson, editors, 2010. Kurt Gödel: Essays for his Centennial. Association for Symbolic Logic and Cambridge University Press.
Qi Feng, Menachem Magidor, and W. Hugh Woodin, 1992. Universally Baire sets of reals. In H. Judah, W. Just, and W. H. Woodin (editors), Set Theory of the Continuum, MSRI publications 26. Springer-Verlag, Berlin.
P. Finsler, 1926. Formale Beweise und die Entscheidbarkeit, Math. Z. 25, 676–682.
J. Folina, 2010. Poincaré’s philosophy of mathematics, Internet Encyclopedia of Philosophy.
M. Foucault, 1970. The Order of Things: An Archaeology of the Human Sciences (a translation of Les Mots et les Choses). Random House, New York.
Curtis Franks, 2009. The Autonomy of Mathematical Knowledge. Cambridge University Press.
Curtis Franks, 2010. Cut as consequence, History Philos. Logic 31(4), 349–379.
G. Frege, 1980. Philosophical and Mathematical Correspondence, edited by G. Gabrielet al. Blackwell Publishers, Oxford.
G. Frege, 1884. Die Grundlagen der Arithmetik, 1884; references to The Foundations of Arithmetic, translated by J. L. Austin. Northwestern University Press, Evanston, IL, 1980.
Chris Freiling, 1986. Axioms of symmetry: throwing darts at the real number line, J. Symbolic Logic 51, 190–200.
M. Friedman, 1999. Reconsidering Logical Positivism. Cambridge University Press.
M. Friedman, 2001. Dynamics of Reason. Cambridge University Press.
M. Friedman, 2010. The a priori in physical theory (presented by Nick Huggett), Central Division Meeting, American Philosophical Association, Chicago, IL.
Pietro Galliani, 2012. Inclusion and exclusion dependencies in team semantics—on some logics of imperfect information, Ann. Pure Appl. Logic 163(1), 68–84.
David Gamarnik, 2002. On deciding stability of constrained homogeneous random walks and queueing systems, Math. Oper. Res. 27(2), 272–293.
David Gamarnik, 2007. On the undecidability of computing stationary distributions and large deviation rates for constrained random walks, Math. Oper. Res. 32(2), 257–265.
Robin Gandy, 1988. The confluence of ideas in 1936. In The Universal Turing Machine: a half-century survey, pp. 55–111. Oxford University Press, New York.
G. Gentzen, 1932. Über die Existenz unabhangiger Axiomensysteme zu unendlichen Satzsystemen, Math. Ann. 107, 329–350; translated as On the existence of independent axiomsystems for infinite sentence systems, in Szabo (1969), pp. 29–52.
G. Gentzen, 1934–1935. Untersuchungen űber das logische Schliessen, Doctoral thesis, University of Gottingen; translated as Investigations into logical deduction, in Szabo (1969), pp. 68–131.
R. George, 1971. Editor’s introduction. In B. Bolzano, Theory of Science. University of California Press, Los Angeles, CA.
J. Gergonne, 1826–1827. Géométrie de situation, Ann. Math. Pures Appl. 17, 214–252.
Victoria Gitman and Joel Hamkins, 2010. A natural model of the multiverse axioms, Notre Dame J. Formal Logic 51(4), 475–484.
Kurt Gödel, 1929. Über die Vollständigkeit des Logikkalküls, Doctoral thesis, University of Vienna; translated by S. Bauer-Mengelberg and Jean van Heijenoort, On the completeness of the calculus of logic, reprinted in Gödel (1986), pp. 60–101.
Kurt Gödel, 1930a. Über die Vollständigkeit des Logikkalküls, Naturwissenschaften 18, 1068; reprinted with English translation in Gödel (1986).
Kurt Gödel, 1930b. Einige metamathematische Resultate über Entscheidungsdefinitheit und Widerspruchsfreiheit, Anz. Akad. Wiss. Wien 67, 214–215; reprinted with English translation in Gödel (1986).
Kurt Gödel, 1930c. Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatsh. Math. Phys. 37(1), 349–360; reprinted with English translation in Gödel (1986), pp. 103–123.
Kurt Gödel, 1931a. Diskussion zur Grundlegung der Mathematik, Erkenntnis 2, 147–151; reprinted with English translation in Gödel (1986).
Kurt Gödel, 1931b. Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I, Monatsh. Math. Phys. 38, 173–198; reprinted with English translation in Gödel (1986).
Kurt Gödel, 1940. The Consistency of the Continuum Hypothesis, Annals of Mathematics Studies, number 3. Princeton University Press, Princeton, NJ.
Kurt Gödel, 1944. Russell’s mathematical logic. In P. A. Schilpp (editor), The Philosophy of Bertrand Russell, pp. 123–153. Northwestern University, Evanstown, IL. Reprinted in Benacerraf and Putnam (1983), pp. 447–469.
Kurt Gödel, 1946. Remarks before the Princeton Bicentennial Conference of problems in mathematics, 1946, lecture first published in Davis (1965), and in Gödel (1990).
Kurt Gödel, 1947. What is Cantor’s continuum problem?Am. Math. Monthly 54, 515–525; reprinted in Benacerraf and Putnam (1983), pp. 470–485.
Kurt Gödel, 1949a. An example of a new type of cosmological solutions of Einstein’s field equations of gravitation, Rev. Mod. Phys. 21, 447–450.
Kurt Gödel, 1949b. A remark about the relationship between relativity theory and idealistic philosophy. In P. Schilpp (editor), Albert Einstein: Philosopher Scientist, pp. 555–562. Library of Living Philosophers, Evanston, IL.
Kurt Gödel, 1961. The modern development of the foundations of mathematics in the light of philosophy, lecture published in Gödel (1995), pp. 374–387.
Kurt Gödel, 1986. Collected Works, volume I, Publications 1929–1936, edited by S. Feferman, J. W. Dawson, Jr., S. C. Kleene, G. H. Moore, R. N. Solovay, and J. van Heijenoort. Oxford University Press, Oxford.
Kurt Gödel, 1990. Collected Works, volume II, Publications 1938–1974, edited by S. Feferman, J. W. Dawson, Jr., S. C. Kleene, G. H. Moore, R. N. Solovay, and J. van Heijenoort. Oxford University Press, Oxford.
Kurt Gödel, 1995. Collected Works, volume III, Unpublished Essays and Lectures, edited by S. Feferman, J. W. Dawson, Jr., W. Goldfarb, C. Parsons, and R. N. Solovay. Oxford University Press, Oxford.
Kurt Gödel, 2003a. Collected Works, volume IV, Correspondence A–G, edited by S. Feferman, J. W. Dawson, Jr., W. Goldfarb, C. Parsons, and W. Sieg. Oxford University Press, Oxford.
Kurt Gödel, 2003b. Collected Works, volume V, Correspondence H–Z, edited by S. Feferman, J. W. Dawson, Jr., W. Goldfarb, C. Parsons, and W. Sieg. Oxford University Press, Oxford.
A. W. Goodman, 1959. On sets of acquaintances and strangers at any party, Am. Math. Monthly 66, 778–783.
R. L. Goodstein, 1953. On the Metamathematics of Algebra by A. Robinson, Mathematical Gazette 37(321), 224–226.
Erich Grädel and Jouko Väänänen, 2013. Dependence and independence, Studia Logica 101(2), 233–236.
Jacques Hadamard, 1945. The Psychology of Invention in the Mathematical Field. Dover, New York.
Hans Hahn, 1956. The crisis in intuition. In James R. Newman (editor), The World of Mathematics, volume III, 1956–1976. Simon & Schuster, New York.
Wolfgang Haken, 1961. Theorie der Normalflächen, Acta Mathematica 105, 245–375 (in German).
Vesa Halava and Tero Harju, 2001. Mortality in matrix semigroups, Am. Math. Monthly 108(7), 649–653.
Vesa Halava, Tero Harju, Mika Hirvensalo, and Juhani Karhumäki, 2005. Skolem’s problem – on the border between decidability and undecidability, TUCS Technical Reports, no. 683, Turku Centre for Computer Science, April 2005. Available at:
Vesa Halava, Tero Harju, and Mika Hirvensalo, 2007. Undecidability bounds for integer matrices using Claus instances, Int. J. Found. Comput. Sci. 18(5), 931–948.
M. Hallett, 2006. Gödel, realism and mathematical “intuition”. In E. Carson and R. Huber (editors), Intuition and the Axiomatic Method, Western Ontario Series in Philosophy of Science, pp. 113–131. Springer, Dordrecht.
Joel Hamkins, 2011. The set-theoretic multiverse: a natural context for set theory (mathematical logic and its applications), Ann. Jpn. Assoc. Philos. Sci. 19, 37–55.
Joel Hamkins, Jonas Reitz, and W. Hugh Woodin, 2007. The ground axiom is consistent with V=HOD, Proc. Am. Math. Soc. I.
Michael Harris, to appear. Not Merely Good, True, and Beautiful. Princeton University Press, Princeton, NJ.
Hamed Hatami and Serguei Norine, 2011. Undecidability of linear inequalities in graph homomorphism densities, J. Am. Math. Soc. 24(2), 547–565.
Kai Hauser, 2006. Gödel’s program revisited, part I: the turn to phenomenology, Bull. Symbolic Logic 12, 529–590.
Robert Aubrey Hearn, 2006. Games, puzzles, and computation, Ph.D. thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. Available at∼bob/hearn-thesis-final.pdf.
Robert A. Hearn and Erik D. Demaine, 2009. Games, Puzzles, and Computation. A K Peters, Wellesley, MA.
Geoffrey Hemion, 1979. On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds, Acta Mathematica 142(1–2), 123–155.
L. Henkin, 1961. Some remarks on infinitely long formulas. In Infinitistic Methods (Proc. Symp. Foundations of Mathematics, Warsaw, 1959), pp. 167–183. Pergamon, Oxford.
P. Hertz, 1929. Über Axiomensysteme für beliebige Satzsysteme, Math. Ann. 101, 457–514.
A. Heyting, 1963. Axiomatic Projective Geometry. Noordhoff, Groningen.
A. Heyting, 1966. Axiomatic method and intuitionism. In Y. Bar-Hillel (editor), Essays on the Foundations of Mathematics: Dedicated to A. A. Fraenkel on his seventieth anniversary, 2nd edition, pp. 237–247. Magnum Press, Jerusalem.
G. Higman, 1961. Subgroups of finitely presented groups, Proc. R. Soc. London, Ser. A 262, 455–475.
D. Hilbert, 1900a. Mathematische Probleme, Göttinger Nachr. 253–297; translated by M. W. Newson in Bull. Am. Math. Soc., 8, 1902, 437–479.
D. Hilbert, 1900b. Über den Zahlbegriff, Jahresber. Deutsch. Math. Verein. 8, 180–183; reprinted in Hilbert (1913).
D. Hilbert, 1913. Grundlagen der Geometrie, 4th edition. Teubner, Leipzig.
D. Hilbert, 1930a. Königsberg radio address. Available at:
D. Hilbert, 1930b. Probleme der Grundlegung der Mathematik, Math. Ann. 102, 1–9.
D. Hilbert and W. Ackermann, 1928. Grundzüge der Theoretischen Logik. Springer, Berlin; translation in Luce (1950).
M. Holz, K. Steffens, and E. Weitz, 1999. Introduction to Cardinal Arithmetic, Birkhäuser Advanced Texts. Birkhäuser, Basel.
David Hume, 1751. Enquiry Concerning the Principles of Morals, §III, part II, ¶ 10, edited by J. Schneewind. Hackett, Indianapolis, IN, 1983.
E. Huntington, 1911. The fundamental propositions of algebra. In J. W. Young (editor), Lectures on Fundamental Concepts of Algebra and Geometry, Chapter IV, pp. 151–210. Macmillan, New York.
Tapani Hyttinen and Saharon Shelah, 1994. Constructing strongly equivalent nonisomorphic models for unsuperstable theories, part A, J. Symbolic Logic 59, 984–996.
Tapani Hyttinen and Heikki Tuuri, 1991. Constructing strongly equivalent nonisomorphic models for unstable theories, Ann. Pure Appl. Logic 52, 203–248.
Tapani Hyttinen, Saharon Shelah, and Heikki Tuuri, 1993. Remarks on strong nonstructure theorems, Notre Dame J. Formal Logic 34, 157–168.
Tapani Hyttinen, Kaisa Kangas, and Jouko Väänänen, 2013. On second order characterizability, Logic J. IGPL 21, 767–787.
Gérard Jacob, 1977–1978. Un algorithme calculant le cardinal, fini ou infini, des demi-groupes de matrices, Theor. Comput. Sci. 5(2), 183–204 (in French, with English summary).
Gérard Jacob, 1978. La finitude des représentations linéaires des semi-groupes est décidable, J. Algebra 52(2), 437–459 (in French, with English summary).
S. Jaśkowski, 1954. Example of a class of systems of ordinary differential equations having no decision method for existence problems, Bull. Acad. Polon. Sci. Cl. III 2, 155–157.
J. P. Jones, 1982. Some undecidable determined games, Int. J. Game Theory 11(2), 63–70.
Lázló Kalmár, 1928–1929. Zur Theorie der abstrakten Spiele, Acta Sci. Math. Szeged 4, 65–85; English translation, The Foundations of Game Theory, volume I, 1997, pp. 247–262.
I. Kant, 1783. Prolegomena to any Future Metaphysics, translated by Paul Carus and James Ellington. Hackett, Indianapolis, IN, 1977.
I. Kant, 1787. Critique of Pure Reason, translated by Norman Kemp Smith. St Martin’s Press, New York, 1965.
I. Kant, 1992. The Jäsche logic. In J. M. Young, editor, Lectures on Logic. Cambridge University Press.
Jarkko Kari, 1996. A small aperiodic set of Wang tiles, Discrete Math. 160(1–3), 259–264.
Jakob Kellner and Saharon Shelah, 2009. Decisive creatures and large continuum, J. Symbolic Logic 74, 73–104.
Juliette Kennedy, 2011. Gödel’s thesis: an appreciation. In Mathias Baaz (editor), Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press, Cambridge.
Juliette Kennedy, 2013. On formalism freeness: implementing Gödel’s 1946 Princeton bicentennial lecture, Bull. Symbolic Logic 19, 351–393.
K. H. Kim and F. W. Roush, 1992. Diophantine undecidability of (t1, t2), J. Algebra 150(1), 35–44.
David A. Klarner, Jean-Camille Birget, and Wade Satterfield, 1991. On the undecidability of the freeness of integer matrix semigroups, Int. J. Algebra Comput. 1(2), 223–226.
S. C. Kleene, 1936. General recursive functions of natural numbers, Math. Ann. 112(1), 727–742.
M. Kline, 1972. Mathematical Thought from Ancient to Modern Times. Oxford University Press, New York.
G. Kneebone, 1963. Mathematical Logic and the Foundations of Mathematics. D. van Nostrand, London.
Peter Koellner, 2006. On the question of absolute undecidability, Philos. Math. 3 (14), 153–188; reprinted with revisions in Feferman et al. (2010).
Peter Koellner, 2009. On reflection principles, Ann. Pure Appl. Logic 157(2–3), 206–219.
Peter Koellner and W. Hugh Woodin, 2010. Large cardinals from determinacy. In M. Foreman and A. Kanamori (editors), Handbook of Set Theory, volume 3, pp. 1951–2119. Springer, Berlin.
Jochen Koenigsmann, 2010. Defining in . Preprint, arXiv:1011.3424.
János Kollár, 2008. Diophantine subsets of function fields of curves, Algebra Number Theory 2(3), 299–311.
Dénes König, 1927. Über eine Schlussweise aus dem Endlichen ins Unendliche, Acta Sci. Math. Szeged 3, 121–130 (in German).
Juha Kontinen and Jouko Väänänen, 2013. Axiomatizing first order consequences in dependence logic. Ann Pure Appl. Logic 164, 1101–1117.
V. Krebs and W. Day, editors, 2010. Seeing Wittgenstein Anew: New Essays on Aspect Seeing. Cambridge University Press, Cambridge.
Georg Kreisel, 1967. Informal rigour and completeness proofs. In Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, edited by Imre Lakatos, volume 1, pp 138–157. North-Holland, Amsterdam.
Georg Kreisel, 1983. Hilbert’s programme. In Paul Benacerraf and Hilary Putnam, editors, Philosophy of Mathematics: Selected Readings, 2nd edition, pp. 207–238. Cambridge University Press.
Georg Kreisel, 1987. Gödel’s excursions into intuitionistic logic. In Gödel Remembered (Salzburg, 1983), History of Logic, IV, pp. 65–186. Bibliopolis, Naples.
S. Kripke, 1980. Naming and Necessity. Harvard University Press, Cambridge, MA.
Stuart A. Kurtz and Janos Simon, 2007. The Undecidability of the Generalized Collatz Problem, Theory and Applications of Models of Computation, Lecture Notes in Computer Science, volume 4484, pp. 542–553. Springer, Berlin.
M. Laczkovich, 2003. The removal of π from some undecidable problems involving elementary functions, Proc. Am. Math. Soc. 131(7), 2235–2240.
Jeffrey C. Lagarias, 1985. The 3x+1 problem and its generalizations, Am. Math. Monthly 92(1), 3–23.
Jeffrey C. Lagarias, editor, 2010. The Ultimate Challenge: the 3x + 1 Problem. American Mathematical Society, Providence, RI.
J. L. Lagrange, 1788. Mécanique Analytique. Desaint, Paris.
J. L. Lagrange, 1797. Théorie des Fonctions Analytiques. Imprimerie de la Republique, Paris.
Serge Lang, 1966. Introduction to Transcendental Numbers. Addison-Wesley, Reading, MA.
C. H. Langford, 1927. On a type of completeness characterizing the general laws for separation of point-pairs, Trans. Am. Math. Soc. 29, 96–110.
P. S. Laplace, 1813. Exposition du Système du Monde. Mme ve Courcier, Paris.
Paul Larson, 2004. The Stationary Tower: Notes on a Course by W. Hugh Woodin. American Mathematical Society, Providence, RI.
Paul Larson and Saharon Shelah, 2009. Splitting stationary sets from weak forms of choice, Math. Logic Q. 55, 299–306.
Richard Laver, 2007. Certain very large cardinals are not created in small forcing extensions, Ann. Pure Appl. Logic 149(1–3), 1–6.
F. W. Lawvere, 1964. An elementary theory of the category of sets, Proc. Natl. Acad. Sci. USA 52, 1506–1511.
A. Lévy, 1960–1961. Principles of reflection in axiomatic set theory, Fundam. Mathematicae 49, 1–10.
Azriel Levy and Robert Solovay, 1967. Measurable cardinals and the continuum hypothesis, Isr. J. Math. 5, 233–248.
Per Lindström, 2000. Quasi-realism in mathematics, Monist 83(1), 122–149.
Joseph Liouville, 1833. Premier et second mémoire sur la détermination des intégrales dont la valeur est algébrique, J. l’École Polytech. 14, 124–148 and 149–193.
Joseph Liouville, 1835. Mémoire sur l’intégration d’une classe de fonctions transcendentes, J. reine angew. Math. 13(2), 93–118.
L. Löwenheim, 1915. Über Möglichkeiten im Relativkalkül, Math. Ann. 76, 447–470.
Robert E. Luce, editor, 1950. Principles of Mathematical Logic. Steinhardt, Chelsea, NY.
Angus Macintyre and A. J. Wilkie, 1996. On the Decidability of the Real Exponential Field, pp. 441–467. Kreiseliana, A. K. Peters, Wellesley, MA.
Penelope Maddy, 1990. Realism in Mathematics. Oxford University Press, Oxford.
Penelope Maddy, 1997. Naturalism in Mathematics. Clarendon Press, Oxford.
Penelope Maddy, 1998a. V = L and MAXIMIZE. In Logic Colloquium 1995 (Haifa), volume 11 of Lecture Notes in Logic, pp. 134–152. Springer, Berlin.
Penelope Maddy, 1998b. Believing the axioms, J. Symbolic Logic 53, 481–511 and 736–764.
Penelope Maddy, 2009. Second Philosophy. Oxford University Press, Oxford.
Penelope Maddy, 2011. Defending the Axioms. Oxford University Press, Oxford.
Arnaldo Mandel and Imre Simon, 1977–1978. On finite semigroups of matrices, Theoret. Comput. Sci. 5(2), 101–111.
Benoît Mandelbrot, 1977. The Fractal Geometry of Nature. W. H. Freeman, New York.
Maurice Margenstern, 2008. The domino problem of the hyperbolic plane is undecidable, Theoret. Comput. Sci. 407(1–3), 29–84.
A. Markov, 1947. The impossibility of certain algorithms in the theory of associative systems, II, Dokl. Akad. Nauk SSSR (N.S.) 58, 353–356 (in Russian).
A. Markov, 1951. The impossibility of certain algorithms in the theory of associative systems, Dokl. Akad. Nauk SSSR (N.S.) 77, 19–20 (in Russian).
A. Markov, 1958. The insolubility of the problem of homeomorphy, Dokl. Akad. Nauk SSSR 121, 218–220 (in Russian).
Donald Martin, 1976. Hilbert’s first problem: the continuum hypothesis. In F. Browder (editor), Mathematical Developments Arising from Hilbert Problems, Proceedings of Symposia in Pure Mathematics, volume 28, pp. 81–92. American Mathematical Society, Providence, RI.
Donald Martin, 1998. Mathematical evidence. In H. G. Dales and G. Oliveri (editors), Truth in Mathematics, pp. 215–231. Clarendon Press, Oxford.
Donald Martin, 2005. Gödel’s conceptual realism, Bull. Symbolic Logic 2, 207–224.
Donald Martin and Robert Solovay, 1970. Internal Cohen extensions, Ann. Math. Logic 2, 143–178.
Yuri V. Matiyasevich, 1970. The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191, 279–282 (in Russian).
Yuri V. Matiyasevich, 1993. Hilbert’s Tenth Problem, Foundations of Computing Series. MIT Press, Cambridge, MA.
Hideyuki Matsumura, 1963. On algebraic groups of birational transformations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 34(8), 151–155.
B. Mazur, 1994. Questions of decidability and undecidability in number theory, J. Symbolic Logic 59(2), 353–371.
Barry Mazur and Karl Rubin, 2010. Ranks of twists of elliptic curves and Hilbert’s tenth problem, Invent. Math. 181, 541–575.
Kenneth McAloon, 1970–1971. Consistency results about ordinal definability, Ann. Math. Logic 2(4), 449–467.
G. S. Mendick and J. K. Truss, 2003. A notion of rank in set theory without choice, Arch. Math. Logic 42(2), 165–178.
Charles F. Miller III, 1992. Decision Problems for Groups – Survey and Reflections, Algorithms and Classification in Combinatorial Group Theory (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., volume 23, pp. 1–59. Springer, New York.
D. Mirimanoff, 1917. Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles, Enseign. Math. 19, 37–52.
William Mitchell, 1979. Hypermeasurable cardinals. In Maurice Boffa, Dirk van Dalen, and Kenneth McAloon (editors), Logic Colloquium ’78, Proceedings of the Colloquium held in Mons, August 1978, pp. 303–316.
Cristopher Moore, 1990. Unpredictability and undecidability in dynamical systems, Phys. Rev. Lett. 64(20), 2354–2357.
Gregory H. Moore, 1988. The emergence of first-order logic. In William Aspray and Philip Kitcher (editors), History and Philosophy of Modern Mathematics, Minnesota Studies in the Philosophy of Science, volume 11, pp. 95–135. University of Minnesota Press, Minneapolis, MN.
Justin Tatch Moore, 2013. Forcing axioms and the continuum hypothesis, part II: transcending ω1-sequences of reals, Acta Mathematica 210, 173–183.
Laurent Moret-Bailly, 2005. Elliptic curves and Hilbert’s tenth problem for algebraic function fields over real and p-adic fields, J. reine angew. Math. 587, 77–143.
Yiannis Moschovakis, 1981. Ordinal games. In A. S. Kechris, D. A. Martin, and Y. N. Moscovakis (editors), Cabal Seminar 77–79, Lecture Notes in Mathematics, volume 839, pp. 169–201. Springer-Verlag, Berlin.
Yiannis Moschovakis, 2009. Descriptive Set Theory, 2nd edition. American Mathematical Society, Providence, RI.
Andrzej Mostowski, 1964. Widerspruchsfreiheit und Unabhängigkeit der Kontinuumhypothese, Elem. Math. 19, 121–125.
John Myhill and Dana Scott, 1971. Ordinal definability. In D. Scott (editor), Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, volume 13, part 1, pp. 271–278. American Mathematical Society, Providence, RI.
Alexander Nabutovsky and Shmuel Weinberger, 1996. Algorithmic unsolvability of the triviality problem for multidimensional knots, Comment. Math. Helv. 71(3), 426–434.
Itay Neeman, 2007. Games of length ω1, J. Math. Logic, 7, 831–824.
P. S. Novikov, 1954. Unsolvability of the conjugacy problem in the theory of groups, Izv. Akad. Nauk SSSR. Ser. Mat. 18, 485–524 (in Russian).
P. S. Novikov, 1955. Ob algoritmičeskoĭ nerazrešimosti problemy toždestva slov v teorii grupp, Tr. Mat. Inst. im. Steklov., Izdat. Akad. Nauk SSSR, 44 (in Russian); English translation, On the algorithmic insolvability of the word problem in group theory, Am. Math. Soc. Transl. 9(2), 1958.
R. J. Nunke, 1977. Whitehead’s problem. In Abelian Group Theory (Proc. Second New Mexico State Univ. Conf.), volume 616 of Lecture Notes in Mathematics, pp. 240–250. Springer, Berlin.
John C. Oxtoby, 1971. Measure and Category. Springer, Berlin.
Pappus, 1941. The treasury of analysis. In I. Thomas (editor), Selections Illustrating the History of Greek Mathematics, volume II, From Aristarchus to Pappus, pp. 596–599. Loeb Classical Library, Harvard University Press, Cambridge, MA.
Jennifer Park, 2012. A universal first order formula defining the ring of integers in a number field, February 28. Preprint, arXiv:1202.6371v1.
Charles Parsons, 1977. What is the iterative conception of set? In R. E. Butts and Jaako Hintikka (editors), Logic, Foundations of Mathematics and Computability Theory (Proc. Fifth Int. Congress on Logic, Methodology and Philosophy of Science, University of Western Ontario, 1975), Part I, University of Western Ontario Series on the Philosophy of Science, volume 9, pp. 335–367. Reidel, Dordrecht.
Charles Parsons, 1980. Mathematical intuition, Proc. Aristotelian Soc., pp. 142–168.
Charles Parsons, 1983. Mathematics in Philosophy. Cornell University Press, Ithaca, NY.
Charles Parsons, 1992. The impredicativity of induction. In Michael Detlefsen (editor), Proof, Logic, and Formalization. Routledge, London.
Charles Parsons, 1995a. Platonism and mathematical intuition in Kurt Gödel’s thought, Bull. Symbolic Logic 1, 44–74.
Charles Parsons, 1995b. Quine and Gödel on analyticity. In Paolo Leonardi and Marco Santambrogio (editors), On Quine: New Essays, pp. 297–313. Cambridge University Press.
Charles Parsons, 1998. Hao Wang as philosopher and interpreter of Gödel, Philos. Math. 3 6(3–24), 9–11; reprinted in Parsons (forthcoming).
Charles Parsons, 2008. Mathematical Thought and its Objects. Cambridge University Press.
Charles Parsons, 2010. Gödel and philosophical idealism, Philos. Math. 18, 166–192.
Charles Parsons, 2014. Philosophy of Mathematics in the Twentieth Century. Harvard University Press, Cambridge, MA.
Michael S. Paterson, 1970. Unsolvability in 3 × 3 matrices, Stud. Appl. Math. 49, 105–107.
C. S. Peirce, 1898. The logic of mathematics in relation to education, Educ. Rev. 15, 209–216.
Thanases Pheidas, 1988. Hilbert’s tenth problem for a class of rings of algebraic integers, Proc. Am. Math. Soc. 104(2), 611–620.
S. Pinker, 1994. The Language Instinct. William Morrow, New York.
H. Poincaré, 1902. La Science et l’Hypothèse. Flammarion, Paris; translated by G. B. Halsted, Science and Hypothesis, The Science Press, New York, 1905.
H. Poincaré, 1913. Dernières Pensées. Flammarion, Paris; translated by J. W. Bolduc, Mathematics and Science: Last Essays, Dover, New York, 1963.
George Polya, 1954. Mathematics and Plausible Reasoning, volume I, Induction and Analogy in Mathematics, volume II, Patterns of Plausible Inference. Princeton University Press, Princeton, NJ.
Bjorn Poonen, 2002. Using elliptic curves of rank one towards the undecidability of Hilbert’s tenth problem over rings of algebraic integers. In Algorithmic Number Theory, pp. 33–42. Springer, Berlin.
Bjorn Poonen, 2008. Undecidability in number theory, Notices Am. Math. Soc. 55(3), 344–350.
Bjorn Poonen, 2009. Characterizing integers among rational numbers with a universal-existential formula, Am. J. Math. 131(3), 675–682.
Bjorn Poonen, 2011. Automorphisms mapping a point into a subvariety, J. Algebraic Geom. 20(4), 785–794.
Emil. L. Post, 1921. Introduction to a general theory of elementary propositions, Am J. Math. 43, 163–185; reprinted in Van Heijenoort (1967a), pp. 265–282.
Emil L. Post, 1944. Recursively enumerable sets of positive integers and their decision problems, Bull. Am. Math. Soc. 50, 284–316.
Emil L. Post, 1946. A variant of a recursively unsolvable problem, Bull. Am. Math. Soc. 52, 264–268.
Emil L. Post, 1947. Recursive unsolvability of a problem of Thue, J. Symbolic Logic 12, 1–11.
Marian Boykan Pour-El and Ian Richards, 1979. A computable ordinary differential equation which possesses no computable solution, Ann. Math. Logic 17(1–2), 61–90.
Marian Boykan Pour-El and Ian Richards, 1983. Noncomputability in analysis and physics: a complete determination of the class of non-computable linear operators, Adv. Math. 48(1), 44–74.
Mojzesz Presburger, 1929. Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, Comptes Rendus du I congrès de Mathématiciens des Pays Slaves, Warsaw, pp. 92–101 (in German); translated by Dale Jacquette, On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation, Hist. Philos. Logic, 12, 1991, 225–233.
Pavel Pudlák, 1996. On the lengths of proofs of consistency: a survey of results. In Collegium Logicum, volume 2 of Coll. Logicum Ann. Kurt-Gödel-Soc., pp. 65–86. Springer, Vienna.
Michael O. Rabin, 1957. Effective computability of winning strategies. Contributions to the Theory of Games, volume 3, Annals of Mathematics Studies, no. 39, pp. 147–157. Princeton University Press, Princeton, NJ.
Michael O. Rabin, 1958. Recursive unsolvability of group theoretic problems, Ann. Math. 2 67, 172–194.
K. Reidemeister, 1930. Vorlesungen über Grundlagen der Geometrie. Springer, Berlin.
W. N. Reinhardt, 1974. Remarks on reflection principles, large cardinals, and elementary embeddings. In Axiomatic Set Theory, part 2, Proceedings of Symposia in Pure Mathematics, volume 13, part 2, pp. 189–205. American Mathematical Society, Providence, RI.
Jonas Reitz, 2007. The ground axiom, J. Symbolic Logic, 72(4), 1299–1317.
Glenn C. Rhoads, 2005. Planar tilings by polyominoes, polyhexes, and polyiamonds, J. Comput. Appl. Math. 174(2), 329–353.
Daniel Richardson, 1968. Some undecidable problems involving elementary functions of a real variable, J. Symbolic Logic 33, 514–520.
Assaf Rinot, 2007. Aspects of singular cofinality, Contrib. Discrete Math. 2(2), 185–204.
Robert H. Risch, 1970. The solution of the problem of integration in finite terms, Bull. Am. Math. Soc. 76, 605–608.
A. Robinson, 1951. On the Metamathematics of Algebra. North-Holland, Amsterdam.
Julia Robinson, 1949. Definability and decision problems in arithmetic, J. Symbolic Logic 14, 98–114.
Raphael M. Robinson, 1971. Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12, 177–209.
Raphael M. Robinson, 1978. Undecidable tiling problems in the hyperbolic plane, Invent. Math. 44(3), 259–264.
W. Rohloff, 2010. Kantian intuition and the applicability of Geometry, Midwest Philosophy of Mathematics Workshop, University of Notre Dame, October, 2010.
Maxwell Rosenlicht, 1972. Integration in finite terms, Am. Math. Monthly 79, 963–972.
Barkley Rosser, 1936. Extensions of some theorems of Gödel and Church, J. Symbolic Logic 1, 87–91.
Lee A. Rubel, 1981. A universal differential equation, Bull. Am. Math. Soc. (N.S.) 4(3), 345–349.
Lee A. Rubel, 1983. Some research problems about algebraic differential equations, Trans. Am. Math. Soc. 280(1), 43–52.
Lee A. Rubel, 1992. Some research problems about algebraic differential equations. II, Illinois J. Math. 36(4), 659–680.
M. Ruse, 1998. Taking Darwin Seriously. Prometheus Books, Amherst, NY.
B. Russell, 1907. The Regressive Method of Discovering the Premises of Mathematics, lecture to the Cambridge Mathematical Club, March 9, 1907. Printed in D. Lackey (editor), Essays in Analysis: Bertrand Russell. George Braziller, New York, 1973.
Grigor Sargsyan, 2009. A tale of hybrid mice, Ph.D. Thesis, University of California, Berkeley, CA.
P. Schilpp, 1951. The Philosophy of Bertrand Russell, Library of Living Philosophers, volume 5, 3rd edition. Tudor Press, New York.
Joseph R. Schoenfield, 1959. On the independence of the axiom of constructibility, Am. J. Math. 81(3), 537–540.
Dana Scott, 1961. Measurable cardinals and constructible sets, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 9, 521–524.
Paul Seidel, 2008. A biased view of symplectic cohomology, Current Developments in Mathematics, 2006, pp. 211–253. International Press, Somerville, MA.
Saharon Shelah, undated a. A.E.C. with not too many models. Preprint.
Saharon Shelah, undated b. Compactness in ZFC of the quantifier on “complete embedding of BA’s.”
Saharon Shelah, undated c. Historic iteration with ℵε-support, Arch. Math. Logic, in press.
Saharon Shelah, undated d. Incompactness in singular cardinals.
Saharon Shelah, undated e. Model theory for Θ-complete ultrapowers.
Saharon Shelah, undated f. Non-reflection of the bad set for ĭθ[λ] and pcf, Acta Mathematica Hungarica, submitted.
Saharon Shelah, undated g. On complicated models. Preprint.
Saharon Shelah, undated h. PCF with narrow choice or ZF + DC + Ax4. Preprint.
Saharon Shelah, undated i. PCF without choice, Arch. Math. Logic, submitted.
Saharon Shelah, undated j. Pseudo pcf, Isr. J. Math., submitted.
Saharon Shelah, 1975. Categoricity in ℵ1 of sentences in lω1,ω(Q), Isr. J. Math. 20, 127–148.
Saharon Shelah, 1987a. Classification of nonelementary classes. ii. Abstract elementary classes. In J. T. Baldwin, editor, Classification Theory (Proceedings of the USA–Israel Conference on Classification Theory, Chicago, IL, December 1985), volume 1292 of Lecture Notes in Mathematics, pp. 419–497. Springer, Berlin.
Saharon Shelah, 1987b. Universal classes. In J. T. Baldwin, editor, Classification Theory (Proceedings of the USA–Israel Conference on Classification Theory, Chicago, IL, December 1985), volume 1292 of Lecture Notes in Mathematics, pp. 264– 418. Springer, Berlin.
Saharon Shelah, 1990. Classification Theory and the Number of Nonisomorphic Models, volume 92 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam.
Saharon Shelah, 1993a. More on cardinal arithmetic, Arch. Math. Logic 32, 399–428.
Saharon Shelah, 1993b. The future of set theory. In Set Theory of the Reals (Ramat Gan, 1991), volume 6 of Israel Math. Conf. Proc., pp. 1–12. Bar-Ilan University, Ramat Gan.
Saharon Shelah, 1994a. Bounding pp(µ) when cf(µ) >µ> ℵ0 using ranks and normal ideals. In Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Oxford University Press.
Saharon Shelah, 1994b. Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Oxford University Press.
Saharon Shelah, 1996. Further cardinal arithmetic, Isr. J. Math. 95, 61–114.
Saharon Shelah, 1997a. The pcf-theorem revisited. In Ronald L. Graham and Jaroslav Nešetřil (editors), The Mathematics of Paul Erdős, II, volume 14 of Algorithms and Combinatorics, pp. 420–459. Springer, Berlin.
Saharon Shelah, 1997b. Set theory without choice: not everything on cofinality is possible, Arch. Math. Logic 36, 81–125. A special volume dedicated to Professor Azriel Levy.
Saharon Shelah, 2000a. Applications of pcf theory, J. Symbolic Logic 65, 1624–1674.
Saharon Shelah, 2000b. The generalized continuum hypothesis revisited, Isr. J. Math. 116, 285–321.
Saharon Shelah, 2000c. On what I do not understand (and have something to say), model theory, Math. Japonica 51, 329–377.
Saharon Shelah, 2000d. On what I do not understand (and have something to say:) part I, Fundam. Mathematicae 166, 1–82.
Saharon Shelah, 2000e. Strong covering without squares, Fundam. Mathematicae 166, 87–107.
Saharon Shelah, 2002. Pcf and infinite free subsets in an algebra, Arch. Math. Logic 41, 321–359.
Saharon Shelah, 2003a. Logical dreams, Bull. Am. Math. Soc. (N.S.) 40(2), 203–228.
Saharon Shelah, 2003b. NNR revisited. Preprint.
Saharon Shelah, 2006. More on the revised gch and the black box, Ann. Pure Appl. Logic 140, 133–160.
Saharon Shelah, 2007. Power set modulo small, the singular of uncountable cofinality, J. Symbolic Logic 72, 226–242.
Saharon Shelah, 2008. Theories with Ehrenfeucht-Fraïssé-equivalent non-isomorphic models, Tbilisi Math. J. 1, 133–164.
Saharon Shelah, 2009a. Classification Theory for Abstract Elementary Classes, volume 18 of Studies in Logic: Mathematical logic and foundations. College Publications, London.
Saharon Shelah, 2009b. Classification Theory for Abstract Elementary Classes 2, volume 20 of Studies in Logic: Mathematical logic and foundations. College Publications, London.
Saharon Shelah, 2009c. Model theory without choice: categoricity, J. Symbolic Logic 74, 361–401.
Saharon Shelah, 2010. Large continuum, oracles, Central Eur. J. Math. 8, 213–234.
Saharon Shelah, 2011a. The character spectrum of β(n), Topology and its Applications 158, 2535–2555.
Saharon Shelah, 2011b. Models of expansions of ∞ with no end extensions, Math. Logic Q. 57, 341–365.
Saharon Shelah, 2012. Nice infinitary logics, J. Am. Math. Soc. 25, 395–427.
Saharon Shelah and Jindrich Zapletal, 1999. Canonical models for ℵ1 combinatorics, Ann. Pure Appl. Logic 98, 217–259.
Alexandra Shlapentokh, 1989. Extension of Hilbert’s tenth problem to some algebraic number fields, Commun. Pure Appl. Math. 42(7), 939–962.
Alexandra Shlapentokh, 2007. Hilbert’s Tenth Problem. Diophantine Classes and Extensions to Global Fields, New Mathematical Monographs, volume 7. Cambridge University Press.
Alexandra Shlapentokh, 2008. Elliptic curves retaining their rank in finite extensions and Hilbert’s tenth problem for rings of algebraic numbers, Trans. Am. Math. Soc. 360(7), 3541–3555.
Joseph R. Shoenfield, 1967. Mathematical Logic. Addison-Wesley, Reading, MA.
Wilfried Sieg, 2006a. Gödel on computability, Philos. Math. 14, 189–207.
Wilfried Sieg, 2006b. Step by recursive step: Church’s analysis of effective calculability. In Church’s Thesis after 70 years, volume 1 of Ontos Math. Log., pp. 456–490. Ontos Verlag, Heusenstamm.
Hava T. Siegelmann and Eduardo D. Sontag, 1995. On the computational power of neural nets, J. Comput. System Sci. 50(1), 132–150.
Waclaw Sierpinski, 1956. L’Hypothèse du Continu. Chelsea, New York.
Th. Skolem, 1923. Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre. Akateeminen Kirjakauppa. English translation in Van Heijenoort (1967a).
Th. Skolem, 1934. Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen, volume 8, pp. 163–188 (in German). 8 Skand. Mat.-Kongr., Stockholm.
Stephen Smale, 1961. Generalized Poincaré’s conjecture in dimensions greater than four, Ann. Math. 74(2), 391–406.
Robert I. Soare, 1996. Computability and recursion, Bull. Symbolic Logic 2(3), 284–321.
Robert I. Soare,2004. Computability theory and differential geometry, Bull. Symbolic Logic 10(4), 457–486.
Sashi Mohan Srivastava, 1988. A Course on Borel Sets. Springer, Berlin.
Daniel T. Stallworth and Fred W. Roush, 1997. An undecidable property of definite integrals, Proc. Am. Math. Soc. 125(7), 2147–2148.
Richard Stanley, 2010. Decidability of chess on an infinite board, MathOverflow, July 20. Available at:
John Steel, 2000. Mathematics needs new axioms, Bull. Symbolic Logic, 6(4), 422–433.
John Steel, 2004. Generic absoluteness and the Continuum Problem, Slides for talk given at the Laguna workshop on philosophy and the Continuum Problem. Available at:
John Steel, 2009. The derived model theorem. In S. Barry Cooperet al. (editors), Logic Colloquium 2006, pp. 280–327. Cambridge University Press.
John Steel and W. Hugh Woodin, 2012. HOD as a core model. In A. Kechris, B. Loewe, and J. Steel (editors), Ordinal Definability and Recursion Theory: the Cabal Seminar. Cambridge University Press, Cambridge.
D. Stewart, 1814. Elements of the Philosophy of the Human Mind, Part I (1792), Part II. William Tegg and Company, London.
M. E. Szabo, 1969. The Collected Papers of Gerhard Gentzen. North-Holland, London.
William Tait, 1986. Truth and proof: the Platonism of mathematics, Synthese 69, 341–370.
William Tait, 1998. Foundations of set theory. In H. G. Dales and Gianluigi Oliveri (editors), Truth in Mathematics, pp. 273–290. Clarendon Press, Oxford.
William Tait, 2005. The Provenance of Pure Reason, Logic and Computation in Philosophy. Oxford University Press, New York.
William Tait, 2008. The five questions. In V. F. Hendricks and H. Leitgeb (editors), Philosophy of Mathematics: Five Questions, pp. 249–263. Automatic Press/VIP, Copenhagen.
William Tait, 2010. Gödel on intuition and on Hilbert’s finitism. In S. Feferman, C. Parsons, and S. Simpson (editors), Kurt Gödel: Essays for His Centennial, Lecture Notes in Logic, pp. 88–108. Cambridge University Press.
Alfred Tarski, 1951. A Decision Method for Elementary Algebra and Geometry, 2nd edition. University of California Press, Berkeley, CA.
Alfred Tarski, 1956. Logic, Semantics, Metamathematics, translated by J. H. Woodger. Oxford University Press, New York.
Richard Tieszen, 1998. Gödel’s path from the incompleteness theorems (1931) to phenomenology (1961), Bull. Symbolic Logic 4, 181–203; reprinted in Tieszen (2005).
Richard Tieszen, 2002. Gödel and the intuition of concepts, Synthese 133, 363–391; reprinted in Tieszen (2005).
Richard Tieszen, 2005. Phenomenology, Logic, and the Philosophy of Mathematics. Cambridge University Press, Cambridge.
A. M. Turing, 1936. On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc. 2 42, 230–265. Erratum: Proc. London Math. Soc. 2, 43 (1937) 544–546.
Jouko Väänänen, 2007. Dependence Logic, volume 70 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge.
Jouko Väänänen, 2012. Second order logic or set theory?Bull. Symbolic Logic 18(1), 91–121.
Jouko Väänänen and Wilfrid Hodges, 2010. Dependence of variables construed as an atomic formula, Ann. Pure Appl. Logic 161(6), 817–828.
Mark van Atten and Juliette Kennedy, 2003. On the philosophical development of Kurt Gödel, Bull. Symbolic Logic 9(4), 425–476.
Mark van Atten and Juliette Kennedy, 2009. Gödel’s modernism: on set-theoretic incompleteness, revisited. In Logicism, Intuitionism, and Formalism, volume 341 of Synthese Library, pp. 303–355. Springer, Dordrecht.
J. Van Heijenoort, 1967a. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge, MA.
J. Van Heijenoort, 1967b. Logic as language and logic as calculus, Synthese 17, 324–330.
O. Veblen, 1903. Hilbert’s foundations of geometry, Monist 13, 303–309.
O. Veblen, 1904. A system of axioms for geometry, Bull. Am. Math. Soc. 5, 343–384.
F. Viète, 1983. The Analytic Art: Nine studies in algebra, geometry, and trigonometry from the Opus restitutae mathematicae analyseos, seu, Algebrâ novâ. Kent State University Press, Kent, OH.
I. A. Volodin, V. E. Kuznecov, and A. T. Fomenko, 1974. The problem of the algorithmic discrimination of the standard three-dimensional sphere, Usp. Mat. Nauk 29(5), 71–168 (in Russian). Appendix by S. P. Novikov.
J. von Neumann, 1925. Eine Axiomatisierung der Mengenlehre. J. reine angewand. Math. 154, 219–240; English translation in Van Heijenoort (1967a).
Maxim Vserminov, editor. Hilbert’s tenth problem page. Website created under the supervision of Yuri Matiyasevich,
Hao Wang, 1961. Proving theorems by pattern recognition—II, Bell System Tech. J. 40(1), 1–41.
Hao Wang, 1974. From Mathematics to Philosophy. Routledge and Kegan Paul, London.
Hao Wang, 1977. Large sets. In R. E. Butts and Jaako Hintikka (editors), Logic, Foundations of Mathematics and Computability Theory (Proc. Fifth Int. Congress on Logic, Methodology and Philosophy of Science, University of Western Ontario, 1975), Part I, University of Western Ontario Series on the Philosophy of Science, volume 9, pp. 309–333. Reidel, Dordrecht.
Hao Wang, 1985. Two commandments of analytic empiricism, J. Philos. 82, 449–462.
Hao Wang, 1988. Reflections on Kurt Gödel. MIT Press, Cambridge, MA.
Hao Wang, 1996. A Logical Journey: From Gödel to Philosophy. MIT Press, Cambridge, MA.
Paul S. Wang, 1974. The undecidability of the existence of zeros of real elementary functions, J. Assoc. Comput. Mach. 21, 586–589.
Shmuel Weinberger, 2005. Computers, Rigidity, and Moduli, M. B. Porter Lectures. Princeton University Press, Princeton, NJ.
H. Weyl, 1918. Das Kontinuum. English translation by Stephen Pollard and Thomas Bole, The Continuum. Dover, Mineola, NY, 1994.
A. Whitehead and B. Russell, 1910. Principia Mathematica, three volumes. Cambridge University Press, Cambridge.
Mark Wilson, 2008. Wandering Significance: An Essay on Conceptual Behaviour. Oxford University Press, New York.
W. Hugh Woodin, 1988. Supercompact cardinals, sets of reals, and weakly homogeneous trees, Proc. Natl. Acad. Sci. USA 85(18), 6587–6591.
W. Hugh Woodin, 2001a. The continuum hypothesis. I, Notices Am. Math. Soc. 48(6), 567–576.
W. Hugh Woodin, 2001b. The continuum hypothesis. II, Notices Am. Math. Soc. 48(7), 681–690.
W. Hugh Woodin, 2002. Correction: “The continuum hypothesis. II”, Notices Am. Math. Soc. 49(1), 46.
W. Hugh Woodin, 2010a. The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, revised edition, volume 1 of de Gruyter Series in Logic and its Applications. Walter de Gruyter, Berlin.
W. Hugh Woodin, 2010b. Suitable extender models I, J. Math. Logic 10, 101–341.
W. Hugh Woodin, 2011. The continuum hypothesis, the generic multiverse of sets, and the Ω-conjecture. In J. Kennedy and R. Kossak (editors), Set Theory, Arithmetic, and Foundations of Mathematics, ASL Lecture Notes in Logic, pp. 13–42.Cambridge University Press.
J. W. Young, 1911. Lectures on Fundamental Concepts of Algebra and Geometry. Macmillan, New York.
Ernst Zermelo, 1913. Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, Proc. Fifth Congress Mathematicians (Cambridge 1912), pp. 501–504 (in German); English translation by Ulrich Schwalbe and Paul Walker, Zermelo and the early history of game theory, Games Econ. Behav. 34, 2001, 123–137.
Ernst Zermelo, 1930. Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundam. Mathematicae 16, 29–47.