×

zbMATH — the first resource for mathematics

Cut as consequence. (English) Zbl 1298.03005
Starting from the observation that Herbrand, Skolem, and Gentzen essentially neglected the question of semantic completeness of first-order logic, as analyzed by Gödel in his dissertation, the author provides a fresh look at Gentzen’s original motivations: “What led Gentzen to ignore questions of and theorems about semantic completeness are the same details of his conception of logic that led to his invention of natural deduction and sequent calculi. The differences between how he thought about basic notions like logical consequence and how they are ordinarily understood today are not subtle but fundamental.” (p. 352 f.).

MSC:
03-03 History of mathematical logic and foundations
01A60 History of mathematics in the 20th century
03F03 Proof theory in general (including proof-theoretic semantics)
03F05 Cut-elimination and normal-form theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.2307/3326862
[2] Bernays P., Contributions to Logic and Methodology in Honor of J. M. Bochenski (1965)
[3] Buss S. R., Handbook of Proof Theory (1998)
[4] Buss S. R., Buss pp 1– (1998)
[5] Bynum T. W, Conceptual Notation and Related Articles (1972)
[6] Dreben B., Feferman et al pp 44– (1986)
[7] DOI: 10.2307/2274427 · Zbl 0645.03002
[10] Fenstad J. E, Thoralf Skolem: Selected Works in Logic (1970)
[11] DOI: 10.1017/CBO9780511642098
[12] Frege, G. 1972. ”Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens Halle: L. Nebert”. Edited by: T. W. 101–208. Bynum as Conceptual Notation: a formula language of pure thought modeled upon the formula language of arithmetic in Bynum
[13] Frege G., ZPPK 81 pp 48– (1882)
[14] Frege G., Sitzungsberichte der Jenaischen Gesellschaft für Medicin und Naturwissenschaft, JZN 16 pp 1– (1883)
[15] Frege G., ’The Foundations of Arithmetic’ (1884) · Zbl 0123.24604
[16] DOI: 10.1007/BF01448897 · Zbl 0005.33803
[17] Gentzen, G. 1934–1935. ”’Untersuchungen über das logische Schliessen”’. PhD Thesis, University of Göttingen, Translated as ’Investigations into logical deduction’ inSzabo 1969, 68–131
[18] DOI: 10.1007/BF01565428 · Zbl 0014.38801
[19] Gentzen G., Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, New Series 4 pp 5– (1938)
[20] Gentzen G., Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, New Series 4 pp 19– (1938)
[21] Girard J. Y., Proof Theory and Logical Complexity 1 (1987) · Zbl 0635.03052
[22] Girard J. Y., Computational Logic pp 215– (1999)
[23] DOI: 10.1017/S096012950100336X · Zbl 1051.03045
[24] DOI: 10.2178/bsl/1052669286 · Zbl 1056.03035
[25] Girard J. Y., The Blind Spot (2004)
[26] Gödel, K. ”’Über die Vollständigkeit des Logikkalküls”’. PhD Thesis, University of Vienna. Translation by S. Bauer-Mengelberg and Jean van Heijenoort as ’On the completeness of the calculus of logic’ reprinted inFeferman et al. 1986, 60–101
[27] DOI: 10.1007/BF01696781 · JFM 56.0046.04
[28] Gödel K., Die Naturwissenschaften 18 pp 1068– (1930)
[29] DOI: 10.1007/BF01700692 · Zbl 0002.00101
[30] Goldfarb W., Jacques Herbrand: Logical Writings (1971)
[31] Herbrand, J. 1930. ”’Recherches sur la théorie de la démonstration”’. PhD Thesis, University of Paris. Translated by W. Goldfarb, except pp. 133–188 translated by B. Dreben and J. van Heijenoort, as ’Investigations in proof theory’ inGoldfarb 1971, 44–202
[32] Herbrand J., Annales de l’Université de Paris 6 pp 186– (1931)
[33] DOI: 10.1007/BF01454856 · JFM 55.0627.01
[34] DOI: 10.1007/BF01782335 · JFM 55.0031.01
[35] Hilbert D., Grundzüge der theoretischen Logik (1928)
[36] Johansson I., Compositio Mathematica 4 pp 119– (1937)
[37] Kanger S., Collected Papers of Stig Kanger with Essays on his Life and Work 1 pp 8– (2001)
[38] Mostowski A., Thirty Years of Foundational Studies pp 43– (1965)
[39] DOI: 10.2178/bsl/1182353872
[40] Skolem T., Matematikerkongressen i Helsingfors 4–7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse pp 217– (1923)
[41] Szabo M. E., The Collected Papers of Gerhard Gentzen (1969) · Zbl 0209.30001
[42] Tarski A., Przeglad Filozoficzny 39 pp 58– (1936)
[43] Van Heijenoort J., From Frege to Gödel: A sourcebook in mathematical logic 1879–1931 (1967)
[44] Van Heijenoort J., van Heijenoort 1985a pp 43– (1976)
[45] Van Heijenoort J., van Heijenoort 1985b pp 99– (1985)
[46] Van Heijenoort J., Selected Essays (1985)
[47] DOI: 10.2178/bsl/1208442829 · Zbl 1145.03003
[48] Von Plato J., Handbook of the History of Logic 5 (2009) · Zbl 1216.03018
[49] Wang H., From Mathematics to Philosophy (1974) · Zbl 0554.03002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.