Nonstandard methods and applications in mathematics, edited by Nigel J. Cutland, Mauro di Nasso and David A. Ross, Lecture Notes in Logic, vol. 25. Association for Symbolic Logic, A.K. Peters, Ltd. Wellesley, Massachusetts, 2006. x + 248 pp.

2007 ◽  
Vol 13 (3) ◽  
pp. 372-374
Author(s):  
Alasdair Urquhart
Keyword(s):  
Symbolic Logic ◽  
Lecture Notes

Completeness Before Post: Bernays, Hilbert, and the Development of Propositional Logic

1999 ◽  
Vol 5 (3) ◽  
pp. 331-366 ◽  
Author(s):  
Richard Zach
Keyword(s):  
Mathematical Logic ◽  
Propositional Logic ◽  
Historical Context ◽  
Subsequent Development ◽  
Symbolic Logic ◽  
David Hilbert ◽  
Semantic Completeness ◽  
Lecture Notes ◽  
Truth Value

AbstractSome of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917–1923. The aim of this paper is to describe these results, focussing primarily on propositional logic, and to put them in their historical context. It is argued that truth-value semantics, syntactic (“Post-”) and semantic completeness, decidability, and other results were first obtained by Hilbert and Bernays in 1918, and that Bernays's role in their discovery and the subsequent development of mathematical logic is much greater than has so far been acknowledged.

Jon Barwise and John Schlipf. On recursively saturated models of arithmetic. Model theory and algebra, A memorial tribute to Abraham Robinson, edited by D. H. Saracino and V. B. Weispfenning, Lecture notes in mathematics, vol. 498, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 42–55. - Patrick Cegielski, Kenneth McAloon, and George Wilmers. Modèles récursivement saturés de l'addition et de la multiplication des entiers naturels. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 57–68. - Julia F. Knight. Theories whose resplendent models are homogeneous. Israel journal of mathematics, vol. 42 (1982), pp. 151–161. - Julia Knight and Mark Nadel. Expansions of models and Turing degrees. The journal of symbolic logic, vol. 47 (1982), pp. 587–604. - Julia Knight and Mark Nadel. Models of arithmetic and closed ideals. The journal of symbolic logic, vol. 47 no. 4 (for 1982, pub. 1983), pp. 833–840. - Henryk Kotlarski. On elementary cuts in models of arithmetic. Fundamenta mathematicae, vol. 115 (1983), pp. 27–31. - H. Kotlarski, S. Krajewski, and A. H. Lachlan. Construction of satisfaction classes for nonstandard models. Canadian mathematical bulletin—Bulletin canadien de mathématiques, vol. 24 (1981), pp. 283–293. - A. H. Lachlan. Full satisfaction classes and recursive saturation. Canadian mathematical bulletin—Bulletin canadien de mathématiques, pp. 295–297. - Leonard Lipshitz and Mark Nadel. The additive structure of models of arithmetic. Proceedings of the American Mathematical Society, vol. 68 (1978), pp. 331–336. - Mark Nadel. On a problem of MacDowell and Specker. The journal of symbolic logic, vol. 45 (1980), pp. 612–622. - C. Smoryński. Back-and-forth inside a recursively saturated model of arithmetic. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 273–278. - C. Smoryński and J. Stavi. Cofinal extension preserves recursive saturation. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7,1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 338–345. - George Wilmers. Minimally saturated models. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 370–380.

1987 ◽  
Vol 52 (1) ◽  
pp. 279-284
Author(s):  
J.-P. Ressayre
Keyword(s):  
New York ◽  
Model Theory ◽  
Foundations Of Mathematics ◽  
Symbolic Logic ◽  
Recursive Saturation ◽  
North Holland Publishing ◽  
Lecture Notes ◽  
Recursively Saturated ◽  
Satisfaction Classes ◽  
Models Of Arithmetic

S. Feferman and W. Sieg Inductive definitions and subsystems of analysis. Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, by Wilfried Buchholz, Solomon Feferman, Wolfram Pohlers, and Wilfried Sieg. Lecture notes in mathematics, vol. 897, Springer-Verlag, Berlin, Heidelberg, and New York, 1981, pp. 16–77. - Solomon Feferman and Wilfried Sieg. Proof theoretic equivalences between classical and constructive theories for analysis. Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, by Wilfried Buchholz, Solomon Feferman, Wolfram Pohlers, and Wilfried Sieg. Lecture notes in mathematics, vol. 897, Springer-Verlag, Berlin, Heidelberg, and New York, 1981, pp. 78–142. - Solomon Feferman. Iterated inductive fixed-point theories: application to Hancock's conjecture. Patras logic symposion, Proceedings of the logic symposion held at Patras, Greece, August 18–22, 1980, edited by George Metakides, Studies in logic and the foundations of mathematics, vol. 109, NorthHolland Publishing Company, Amsterdam, New York, and Oxford, 1982, pp. 171–196. - Solomon Feferman. Monotone inductive definitions. The L. E. J. Brouwer centenary symposium, Proceedings of the conference held in Noordwijkerhout, 8–13 June, 1981, edited by A. S. Troelstra and D. van Dalen, Studies in logic and the foundations of mathematics, vol. 110, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1982, pp. 77–89. - S. Feferman and G. Jäger. Choice principles, the bar rule and autonomously iterated comprehension schemes in analysis. The journal of symbolic logic, vol. 48 (1983), pp. 63–70.

1994 ◽  
Vol 59 (2) ◽  
pp. 668-670
Author(s):  
Helmut Pfeiffer
Keyword(s):  
New York ◽  
Fixed Point ◽  
Publishing Company ◽  
Foundations Of Mathematics ◽  
Symbolic Logic ◽  
Theoretical Studies ◽  
North Holland Publishing Company ◽  
Inductive Definitions ◽  
North Holland Publishing ◽  
Lecture Notes
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