Abstract
Work on how to axiomatize the subtheories of a first-order theory in which only a proper subset of their extra-logical vocabulary is being used led to a theorem on recursive axiomatizability and to an interpolation theorem for first-order logic. There were some fortuitous events and several logicians played a helpful role.
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References
Beth E.W. (1953). On Padoa’s method in the theory of definitions. Indagationes Mathematicae 15: 330–339
Büchi R.J. and Craig W. (1956). Notes on the family PC Δ of sets of models. Abstract. Journal of Symbolic Logic 21: 221–223
Craig, W. (1951). A theorem about first order functional calculus with identity. Ph.D. thesis, Harvard, 242 pp.
Craig W. (1953). On axiomatizability within a system. Journal of Symbolic Logic 18: 30–32
Craig W. (1956). Review of E. W. Beth, On Padoa’s method in the theory of definitions. Journal of Symbolic Logic 21: 194–195
Craig W. (1957a). Linear reasoning. A new form of the Herbrand–Gentzen Theorem. Journal of Symbolic Logic 22: 250–268
Craig W. (1957b). Three uses of the Herbrand–Gentzen Theorem in relating proof theory and model theory. Journal of Symbolic Logic 22: 269–285
Craig W. (1958). Replacement of auxiliary expressions. Philosophical Review 67: 38–55
Craig W. (1960). Bases for first-order theories and subtheories. Journal of Symbolic Logic 25: 97–142
Craig W. and Vaught R. (1958). Finite axiomatizability using additional predicates. Journal of Symbolic Logic 25: 289–308
Ebbinghaus, A. D. (1985). Extended logics. The general framework. In J. Barwise & S. Feferman (Eds.), Model-theoretic logics (p. x+893). Springer-Verlag.
Gentzen, G. (1934–1935). Untersuchungen über das logische Schliessen. Mathematicsche Zeitschrift, 39, 179–210, 405–431.
Herbrand, J. (1930) Recherches sur la théorie de la démonstration. Travaux de la Société des Sciences et Lettres de Varsovie III, 128 pp.
Kleene S.C. (1952). Finite axiomatizability of theories in the predicate calculus using additional predicate symbols. Memoirs of the American Mathematical Society 10: 27–68
Robinson A. (1956). A result ones consistency and its application to the theory of definitions. Indagationes Math 18: 17–58
Turing A. (1936). On computable numbers with an application to the Entscheidungs problem. Proceedings of the London Mathematical Society ser. 2 42: 230–265
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Craig, W. The road to two theorems of logic. Synthese 164, 333–339 (2008). https://doi.org/10.1007/s11229-008-9353-3
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DOI: https://doi.org/10.1007/s11229-008-9353-3