Linked bibliography for the SEP article "The Development of Proof Theory" by Jan von Plato |
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Introductions to proof theory:
- Negri, S. and von Plato, J. Structural Proof Theory, Cambridge 2001.
- Takeuti, G. Proof Theory, North-Holland, 2nd ed, 1987.
- Troelstra, A. and Schwichtenberg, H. Basic Proof Theory, Cambridge, 2nd ed. 2000.
- The Handbook of Proof Theory (Sam Buss ed., 1998) contains reviews of specialized topics in proof theory.
Original papers (in order of first publication):
Hilbert: The central papers are found in English translation in Van Heijenoort's From Frege to Gödel, Harvard 1967. Consult, for the influence on Gentzen, also:
- Hilbert, D. Die Grundlegung der elementaren Zahlenlehre, Mathematische Annalen, vol. 104, pp. 484–494, 1931.
Skolem: The papers in logic are found in Selected Works in Logic, ed. Fenstad 1970. Therein, pp. 103–136:
- Skolem, T. Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze, nebst einem Theoreme über dichte Mengen, (original 1920).
Gentzen: The published papers can be found in English translation in The Collected Papers of Gerhard Gentzen ed. Szabo, 1969. The proof of normalization is in:
- Gentzen, G. The normalization of derivations, The Bulletin of Symbolic Logic, vol. 14, pp. 245–257, 2008.
Other original material from Gentzen cited or referred to above is from the biography Logic's Lost Genius: The Life and Work of Gerhard Gentzen, by Eckart Menzler-Trott, American Mathematical Society 2007.
- Gödel. K. Collected Papers I–V, Oxford 1986–2003.
- Jaskowski, S. On the rules of supposition in formal logic, (original 1934), in S. McCall, ed, Polish Logic 1920–1939, pp. 232–258, Oxford 1967.
- Ackermann, W. Zur Widerspruchsfreiheit der Zahlentheorie, Mathematische Annalen, vol. 117, pp. 162–1940, 940.
- Ketonen, O. Untersuchungen zum Prädikatenkalkül, (Annales Acad. Sci. Fenn, Ser. A.I. 23), 1944.
- Schütte, K. Schlussweisen-Kalküle der Prädikatenlogik, Mathematische Annalen, vol. 122, pp. 47–65, 1950.
- Schütte, K. Beweistheoretische Erfassung der unendlichen Induktion in der Zahlentheorie, Mathematische Annalen, vol. 122, pp. 369–389, 1951.
- Kreisel, G. On the interpretation of non-finitist proofs: Part I, The Journal of Symbolic Logic, vol. 16, pp. 241–267, 1951.
- Kleene, S. Introduction to Metamathematics, North-Holland 1952.
- Beth, E. Semantic entailement and formal derivability, Mededelingen der Koninklijke Nederlandse Akademie van wetenschappen, afd. letterkunde, vol. 18, no. 13, 1955.
- Curry, H. Foundations of Mathematical Logic, original 1963, Dover reprint 1977. (Scholar)
- Prawitz, D. Natural Deduction: A Proof-Theoretical Study, 1965, Dover reprint with a new preface 2006. (Scholar)
- Prawitz, D. Ideas and results in proof theory, in J. Fenstad, ed, Proceedings of the Second Scandinavian Logic Symposium, pp. 235–308, North-Holland 1971.
- Howard, W. The formulae-as-types notion of construction, in J. Seldin and J. Hindley, eds, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 480–490, Academic Press 1980.
- Dragalin, A. Mathematical Intuitionism: Introduction to Proof Theory, American Mathematical Society 1988.
- Rathjen, M. Recent advances in ordinal analysis; Π12-CA and related systems, The Bulletin of Symbolic Logic, vol. 1, pp. 468–485, 1995.
Secondary literature:
- Bernays, P. Betrachtungen zum Sequenzen-kalkul, in Contributions to Logic and Methodology in Honor of J. M. Bochenski, pp. 1–44, North-Holland 1965.
- Bernays, P. On the original Gentzen consistency proof for number theory, in J. Myhill et al., eds, Intuitionism and Proof Theory, pp. 409–417, North-Holland 1970.
- Feferman, S. Highlights in proof theory, pp. 11–31 in V. Hendricks et al., eds.
- Hendricks, V. et al., eds, Proof Theory: History and Philosophical Significance, Kluwer 2000.
- von Plato, J. In the shadows of the Löwenheim-Skolem theorem: early combinatorial analyses of mathematical proofs, The Bulletin of Symbolic Logic, vol. 13, pp. 189–225, 2007.
- von Plato, J. From Hilbert's programme to Gentzen's programme (appendix in Menzler-Trott 2007).
- von Plato, J. Gentzen's logic, in Handbook of the History and Philosophy of Logic, vol. 5, Elsevier, in press.
- Smorynski, C. Hilbert's programme, (appendix in Menzler-Trott 2007).
- Tait, W. Gödel's reformulation of Gentzen's first consistency proof for arithmetic: the no-counterexample interpretation, The Bulletin of Symbolic Logic, vol. 11, pp. 225–238, 2005.
- Troelstra, A. and Schwichtenberg, H. Basic Proof Theory, Cambridge, 2nd ed. 2000), the historical notes at the end of each chapter.
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