The Rise of Modern Logic · Hilary 2010

Plan of lectures

  1. Introduction and Frege's Begriffsschrift
  2. Frege's logicism
  3. Russell and type theory
  4. Zermelo's axioms
  5. Set theory between 1908 and 1930
  6. The emergence of first-order logic
  7. Hilbert's programme
  8. Gödel's theorems and their impact on Hilbert's programme

Recommended Readings

Many of the original papers are contained in:

Jan van Heijenoort (ed.) From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Cambridge, MA: Harvard University Press, 1967.

And for those who want to read German the following collection is useful:

Karel Berka and Lothar Kreiser (eds.) Logik-Texte: Kommentierte Auswahl zur Geschichte der modernen Logik, Akademie-Verlag, Berlin, first edition 1971, third, enlarged edition 1983, fourth edition 1986.

For an overview of the perriod from 1900 on see:

Paolo Mancosu, Richard Zach and Calixto Badesa: "The Development of Mathematical Logic from Russell to Tarski: 1900–1935", in Leila Haaparanta, ed., The Development of Modern Logic, New York and Oxford: Oxford University Press, 2005. It is also available-at least now-as a pdf file.

Frege

Gottlob Frege, Begriffsschrift , eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle, 1879; English translation by Stephen Bauer-Mengelberg, "Begriffschrift, a formula language, modeled upon that of arithmetic, for pure thought", in Jean van Heijenoort (ed.), From Frege to Gödel: a source book in mathematical logic 1879-1931, Harvard Unversity Press, 1967, pp. 1-82 (reprinted in Frege and Gödel: two fundamental texts in mathematical logic, Harvard University Press, 1970); English translation by T.W. Bynum, "Conceptual notation, a formula language of pure thought modelled upon the formula language of arithmetic", in T.W. Bynum (ed.), Gottlob Frege Conceptual Notation and related articles, Oxford University Press, 1972, pp. 101-203.

Michael Beaney, 1997, The Frege Reader, Oxford: Blackwell

George Boolos, "Reading the Begriffsschrift", Mind 94 (1985), pp. 331-344; reprinted in William Demopoulos (ed.), Frege's Philosophy of Mathematics, Harvard University Press, 1995, pp. 163-181, and in George Boolos, Logic, Logic, and Logic, Harvard University Press, 1998, pp. 155-170.

Gottlob Frege, Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, Pohle, Jena, vol. 1 1893, vol. 2 1903; English translation by Montgomery Furth of Part I, The Basic Laws of Arithmetic: Exposition of the System, University of California Press, 1964; English translation by various translators in Peter Geach and Max Black (eds), Translations from the Philosophical Writings of Gottlob Frege, Basil Blackwell, Oxford, 1960, pp. 137-244.

Richard G. Heck, "The development of arithmetic in Frege's Grundgesetze der Arithmetik", The Journal of Symbolic Logic 58 (1993), pp. 579-601.

Richard G. Heck, "Definition by induction in Frege's Grundgesetze der Arithmetik", William Demopoulos (ed.) Frege's Philosophy of Mathematics, Harvard University Press, 1995, pp. 295-233.

Richard G. Heck, "The finite and the infinite in Frege's Grundgetze der Arithmetik", Matthias Schirn (ed.) The Philosophy of Mathematics Today, Oxford University Press, 1998, pp. 429-466.

Anthony Kenny, Frege: An Introduction to the Founder of Modern Analytic Philosophy, Penguin Books, Chapters 8 and 9, pp. 142-177.

Gottlob Frege, Grundlagen der Arithmetik, Jena, 1884; English translation by J.L. Austin, The Foundations of Arithmetik, §§62-83, pp. 73-95.

George Boolos, "On the proof of Frege's theorem", Adam Morton and Stephen P. Stich (eds), Benacerraf and his Critics, Blackwell, Oxford, 1996, pp. 143-59; reprinted in George Boolos, Logic, Logic, and Logic, Harvard University Press, 1998, pp.275-290.

George Boolos, "Frege's Theorem and the Peano Postulates", The Bulletin of Symbolic Logic 1 (1995), pp. 317-326; reprinted in George Boolos, Logic, Logic, and Logic, Harvard University Press, 1998, pp.291-300.

Richard G. Heck, "Frege's Theorem: an introduction", The Harvard Review of Philosophy 7 (1999), pp. 56-73.

Edward Zalta, "Frege's Logic, Theorem, and Foundations for Arithmetic", in the Stanford Encyclopedia of Philosophy.

Russell

Bertrand Russell, "The doctrine of types", Appendix B of The Principles of Mathematics, George Allen & Unwin, 1903, pp. 523-528

Bertrand Russell, "Mathematical logic as based on the theory of types", American Jounral of Mathematics 30 (1908), pp. 222-262; reprinted in R.C. Marsh (ed.), Logic and Knowledge: Essays 1901-1950, Allen and Unwin, London, 1956, pp. 59-102, and in Jean van Heijenoort (ed.), From Frege to Gödel, Harvard, 1967, pp. 150-182.

Bertrand Russell, "The Theory of Logical Types", Chapter 2 of the Introduction to Principia Mathematica, Volume I, Cambridge University Press, 1910, 2nd edn 1927, pp. 37-65

Bertrand Russell, Introduction to Mathematical Philosophy, George Allen & Unwin, 1919, Chapters 8, 13, 15, 17, 18, pp. 77-88, 131-143, 155-166, 181-206.

Martin Godwyn and Andrew D. Irvine, "Bertrand Russell's logicism", Nicholas Griffin (ed.), The Cambridge Companion to Bertrand Russell, Cambridge university Press, Cambridge, 2003, pp. 171-201.

Alasdair Urquhart, "The theory of types", Nicholas Griffin (ed.), The Cambridge Companion to Bertrand Russell, Cambridge university Press, Cambridge, 2003, pp. 286-309.

Charles Chihara, "Russell's solution to the paradoxes", Chapter 1 of Ontology and the Vicious Circle Principle, Cornell University Press, 1973, pp. 1-59.

Warren Goldfarb, "Russell's reasons for ramification", C.W. Savage and C. A. Anderson (eds), Rereading Russell, Essays in Bertrand Russell's Metaphysics and Epistemology, University of Minnesota Press, 1989, pp. 24-40.

Georg Kreisel, biographical memoir of Bertrand Russell, Chapter II, "Mathematical logic and logical foundations of mathematics", Biographical Memoirs of Fellows of The Royal Society 19 (1973), pp. 591-606.

Bernard Linsky, "Was the axiom of reducibility a principle of logic?", The Journal of the Bertrand Russell Archive, New Series vol 10, 1990-91, pp. 125-140; reprinted in William W. Tait (ed.), Early Analytic Philosophy: Frege, Russell, Wittgenstein, Open Court Publishing Co., Chicago, 1997, pp. 107-121.

Jules Richard, "Les principes des mathématique et le problème des ensembles", Revue génerale des sciences pures et appliqués 16, p. 541; English translation by Jean van Heijenoort in Jean van Heijenoort (ed.), From Frege to Gödel; a source book in mathematical logic 1879-1931, pp. 142-144.

Warren Goldfarb, "Poincaré against the logicists", W. Aspray and P. Kitcher (eds), History and Philosophy of Modern Mathematics, University of Minnesota Press, 1988, pp. 61-81.

Frank Ramsey, "The foundations of mathematics", Proceedings of the London Mathematical Society Ser 2, Vol 25 (1925), pp. 338-384; reprinted in collections of Ramsey's papers

Rudolf Carnap, "Die logizistische Grundlegung der Mathematik", Erkenntnis 2 [1931], pp. 91-105; English translation by Erna Putnam and Gerald J. Massey, "The logicist foundations of mathematics", Paul Benacerraf and Hilary Putnam (eds), Philosophy of Mathematics Selected Readings, Basil Blackwell, Oxford, 1964, pp. 31-41; 2nd edn Cambridge University Press, 1983, pp. 41-52.

Kurt Gödel, "Russell's mathematical logic", P.A. Schilipp (ed.), The Philosophy of Bertrand Russell, 1944, Northwester University Press, Evanston, Illinois, pp. 125-153; reprinted, with Introductory Note by Charles Parsons, in Kurt Gödel Collected Works Volume II, Oxford University Press, 1990, pp. 102-141.

Hermann Weyl, Das Kontinuum; Kritische Untersuchungen über die Grundlagen der Analysis, Veit, Leipzig, 1918; English translation by Stephen Pollard and Thomas Bole, The Continuum; a Critical Examination of the Foundation of Analysis, The Thomas Jefferson Univesity Press, Kirksville, MO, 1987.

Paolo Mancosu, "Hermann Weyl: Predicativity and an Intuitionistic Excursion", From Brouwer to Hilbert: The debate on the Foundations of Mathematics in the 1920s, Oxford University Press, 1998, pp. 65-85.

Cantor

Georg Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen, Teubner, Leipzig, 1883; English translation by William Ewald, "Foundations of a general theory of manifolds: a mathematico-philosophical investigation into the theory of the infinite", William Ewald (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume II, Oxford University Press, 1996, pp. 878-920,

Georg Cantor, Letter to Richard Dedekind 28 July 1899, English translation by Stefan Bauer Mengelberg and Jean van Heijenoort, Jean van Heijenoort (ed.), From Frege to Gödel: a source book in mathematical logic 1979-1931, Harvard University Press, 1967, pp. 113-117.

Hallett, M. (1984) Cantorian Set Theory and Limitation of Size, Oxford: Clarendon Press. (This book does not only cover Cantor's development of set theory but also later contributions).

Zermelo and the axiomatisation of set theory

Ernst Zermelo, "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65 (1908), pp. 261-281; English translation by Stefan Bauer-Mengelberg, "Investigations in the foundations of set theory I", in Jean van Heijenoort (ed.), From Frege to Gödel: a source book in mathematical logic 1879-1931, Harvard University Press, 1967, pp. 199-215.

Gregory H. Moore, "The origins of Zermelo's axiomatization of set theory", Journal of Philosophical Logic 7 (1978), pp. 307-329.

Gregory H. Moore, "Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory", History and Philosophy of Logic 1 (1980), pp. 95-127.

George Boolos, "The iterative conception of set", Journal of Philosophy 68 (1971), pp. 215-132; reprinted in Benacerraf and Putnam (eds), Philosophy of Mathematics: Selected Readings, second edition, Cambridge University Press, 1983, pp. 486-502; reprinted in George Boolos, Logic, Logic, and Logic, Harvard University Press, 1998, pp. 13-29.

Thoralf Skolem, "Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Matematikerkongressen i Helsingofors den 4-7 Juli 1922, Helsinki, 1923; English translation by Stefan Bauer-Mengelberg, "Some remarks on axiomatized set theory", Jean van Heijenoort (ed.), From Frege to Gödel: A source book in mathematical logic 1879-1931, Harvard University Press, 1967, pp. 290-301.

Ernst Zermelo, “Über den Begriff der Definitheit in der Axiomatik”, Fundamenta Mathematicae 14 (1929), pp. 339-344.

Ernst Zermelo, “Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre”, Fundamenta Mathematicae 16 (1930), pp. 29-47; English translation and introductory note by Michael Hallett, “On boundary numbers and domains of sets: new investigations in the foundations of set theory”, William B. Ewald (ed.) From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume 2, Oxford University Press, 1996, pp. 1208-1233.

Akihiro Kanamori, The Higher Infinite, §1 “Inaccessibility”, Springer, 1997, pp. 16-21.

Akiro Kanamori, "Zermelo and set theory", Bulletin of Symbolic Logic 10 (2004), 487-553.

Penelope Maddy, " Believing the axioms I", Journal of Symbolic Logic 53 (1988), 481-511 (only the first part is relevant for the lectures).

Michael Potter, (1998). "Set theory, different systems of". In E. Craig (Ed.), Routledge Encyclopedia of Philosophy. London: Routledge. Retrieved November 11, 2005, from http://www.rep.routledge.com/article/Y025 This article gives an overview of alternative approaches.

The emergence of first-order logic

Gregory H. Moore, “The emergence of first-order logic”, William Aspray and Philip Kitcher (eds), History and Philosophy of Modern Mathematiacs, Minnesota Studies in the Philosophy of Science vol XI. University of Minnesota Press, 1988, pp. 95-135.

Gregory H. Moore, “Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory”, History and Philosophy of Logic 1 (1980), pp. 95-127.

Stewart Shapiro, Foundations without Foundationalism, Clarendon Press, 1997 (I recommend this book if you are interested in second-order logic from a non-historical point of view; it provides a clear and modern exposition of the metamathematics of second-order logic).

Leopold Löwenheim, “¨Über Möglichkeiten im Relativkalkül”, Mathematische Annalen 76 (1915), pp. 447-470; English translation by Stefan Bauer-Mengelberg, “On possibilities in the calculus of relatives”, Jean van Heijenoort (ed.), From Frege to Gödel: a source book in mathematical logic 1879-1931, Harvard University Press, 1967, pp. 228-251.

Thoralf Skolem, “Logisch-kombinatorsche Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischser Sätze nebst einem Theoreme über dichte Menge”, §1. Vereinfachter Beweis eines Satzes von L. Löwenheim. Verallgemeinerungen des Satzes”,Videnskapsselskapets skrifter, I. Matematisk-naturvidenskablelig klasse, no. 4, 1920; reprinted in Jens Erik Fenstad (ed.), Th. Skolem Selected Works in Logic, Skandinavian University Books, 1970, pp. 103-115; English translation by Stefan Bauer-Mengelberg, “Logico-combinatorial investigations in the satisfiability or provability of mathematical propositions: A simplified proof of a theorem by L. Löwenheim and generalizations of the theorem”, Jean van Heijenoort (ed.) op. cit., pp. 252-263.

Thoralf Skolem, “Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre”, Matematikerkongressen I Helsingfors den 4-7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse, Akademiska Bokhandeln, Helsinki, 1923, pp. 217-232; reprinted in Jens Erik Fenstad (ed.) op. cit., pp. 137-152; English translation by Stefan Bauer-Mengelberg, “Some remarks on axiomatized set theory”, Jean van Heijenoor (ed.), op. cit., pp. 290-301.

Kurt Gödel, “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik 37, 349-360; English translation by Stefan Bauer-Mengelberg, “The completeness of the axioms of the functional calculus of logic”, Jean van Heijenoor (ed.), op. cit., pp. 582-591.

Hilbert's programme

David Hilbert, “Axiomatisches Denken”, Mathematische Annalen 78 (1918), pp. 405-15; English translation by William Ewald, “Axiomatic thought”, in William Ewald (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, 1996, pp. 1105-1115,

David Hilbert, “Über das Unendliche”, Mathemataische Annalen 95 (1926), pp. 161-190; English translation by Stefan Bauer-Mengelberg, “On the infinite”, in Jean van Heijenoort (ed.), From Frege to Gödel: a source book in mathematical logic 1879-1931, Harvard University Press, 1967, pp. 367-392.

David Hilbert, “Die Grundlagen der Mathematik”, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 6 (1928), pp. 65-85; English translation by Stefan Bauer-Mengelberg and Dagfinn Føllesdal, “The foundations of mathematics” in van Heijenoort, op. cit., pp. 464-479.

Detlefsen, Michael, Hilbert's Program, Dordrecht, Reidel, 1986.

Georg Kreisel, “Hilbert’s programme”, Dialectica 12 (1958), pp. 346-372; revised version in Paul Benacerraf & Hilary Putnam (eds), Philosophy of Mathematics: Selected Readings, Prentice-Hall, Englewood Cliffs, New Jersey, 1964, pp. 157-180; the 1964 version is reprinted with a Postscript (1978) in the second edition of Benacerraf & Putnam, op. cit . Cambridge University Press, 1983, pp. 207-238.

Paolo Mancosu, “Hilbert and Bernays on metamathematics”, Paolo Mancosu (ed.) From Brouwer to Hilbert: the Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, 1998, pp. 149-188.

W.W. Tait, “The substitution method”, The Journal of Symbolic Logic 30 (1965), pp. 175-192.

Richard Zach, “Hilbert's program then and now”, in Dale Jacquette, ed., Handbook of the Philosophy of Logic (North-Holland, Amsterdam), 43 pp., forthcoming, pdf file

Gödel's incompleteness theorems and their impact on Hilbert's programme

Kurt Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”Monatshefte fur Mathematik und Physik 38 (1931), pp. 173-198; English translation by Jean van Heijenoort, “On formally undecidable sentences of Principia Mathematica and related systems I”, Jean van Heijenoort (ed.), From Frege to Gödel: a source book in mathematical logic 1979-1931, pp. 596-616; van Heijenoort translation reprinted, with facing German original text, in Solomon Feferman et al (eds), Kurt Gödel Collected Works Volume I, Oxford University Press, 1986, pp. 144-195.

Michael Detlefsen, “On an alleged refutation of Hilbert’s program using Gödel’s first incompleteness theorem”, The Journal of Philosophical Logic 18 (1990), reprinted in Michael Detlefsen (ed.), Proof, Logic and Formalization, Routledge, London, 1992, pp. 199-235.

Paolo Mancosu, “Between Vienna and Berlin: The immediate reception of Gödel’s incompleteness theorems”, History and Philosophy of Logic 20 (1999), pp. 33-45.

Zach, Richard, "Hilbert's Program", The Stanford Encyclopedia of Philosophy (Fall 2003 Edition), Edward N. Zalta (ed.), section 4

Intuitionism

Posy, C. “Intuitionism and Philosophy” in The Oxford Handbook of Philosophy of Mathematics and Logic, Steward Shapiro (ed.), Oxford, 2005, 318–355.

Dummett, M. Elements of Intuitionism, Oxford University Press, 1977

McCarty, David C. (1998). Intuitionism. In E. Craig (Ed.), Routledge Encyclopedia of Philosophy. London: Routledge. Retrieved November 25, 2005, from http://www.rep.routledge.com/article/Y062