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Completeness before Post: Bernays, Hilbert, and the development of propositional logic. (English) Zbl 0942.03003

This paper, based on unpublished and archival sources, gives deep insights into the collaboration between D. Hilbert and his assistant P. Bernays during the formative period of “Hilbert’s program”. The author is most interested in the distinction of syntax and semantics and questions of completeness and decidability in propositional logic. He presents Bernays’s Habilitationsschrift “Beiträge zur axiomatischen Behandlung des Logik-Kalküls” [Bernays papers, library of the ETH Zürich, Hs. 973.192] as elaboration and in several respects original extension of ideas which can be found in lecture courses delivered by Hilbert in 1905 (“Logische Principien des mathematischen Denkens”) and 1917/18 (“Prinzipien der Mathematik”), the last of which is available in Bernays’s notes. Hilbert’s ideas concerning an axiomatized system of propositional logic with consistency proof and a completeness proof applying the normal form theorem, were used by Bernays who filled in “the last gaps between these remarks and a completely modern presentation of propositional logic” (p.341). By seriously examining Bernays shorthand notes of Hilbert’s 1917/18 lecture course, his elaboration of this course and his Habilitationsschrift, the author gives evidence, however, that Bernays was much more than an executor of Hilbertian ideas. The author regards as Bernays’s and Hilbert’s most important advances in the development of mathematical logic and the foundations of mathematics “the distinction between syntax and semantics, the formulation of syntactic and semantic completeness, the proof of completeness for the propositional calculus, and the proof of decidability” (p.353).

MSC:

03-03 History of mathematical logic and foundations
01A60 History of mathematics in the 20th century
03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics
03B25 Decidability of theories and sets of sentences
03B30 Foundations of classical theories (including reverse mathematics)

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