Abstract
This paper initiates an investigation which seeks to explain elementary definability as the classical results of mathematical logic (the completeness, compactness and Löwenheim-Skolem theorems) explain elementary logical consequence. The theorems of Beth and Svenonius are basic in this approach and introduce automorphism groups as a means of studying these problems. It is shown that for a complete theory T, the definability relation of Beth (or Svenonius) yields an upper semi-lattice whose elements (concepts) are interdefinable formulas of T (formulas having equal automorphism groups in all models of T). It is shown that there are countable models A of T such that two formulae are distinct (not interdefinable) in T if and only if they are distinct (have different automorphism groups) in A. The notion of a concept h being normal in a theory T is introduced. Here the upper semi-lattice of all concepts which define h is proved to be a finite lattice — anti-isomorphic to the lattice of subgroups of the corresponding automorphism group. Connections with the Galois theory of fields are discussed.
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References
E. W. Beth, On Padoa’s method in the theory of definitions, Indag. Math. 15 (1953).
J. R. Buchi, Relatively Categorical and Normal Theories, in The Theory of Models, Amsterdam, 1965.
J. R. Buchi and K. J. Danhof, Model Theoretic Approaches to Definability, Z. Math. Logik Grundlagen Math. 18 (1972), 61–70.
J. R. Buchi and J. B. Wright, The theory of proportionality as an abstraction of group theory, Math. Ann. 130 (1955).
J.R. Buchi and J.B. Wright, Invariants of the anti-automorphisms of a group, Proc. Amer. Math. Soc. 8 (1957).
W. Craig, Linear reasoning. A new form of the Herbrand-Gentzen theorem, J. Symbolic Logic 22 (1957).
W. Craig, Tree uses of the Herband-Gentzen theorem in relating model theorey and proof theory, J. Symbolic Logic 22 (1957).
K.J. Danhof, Concepts in normal theories, Notices Amer. Math. Soc. 71 T-E26 (1971).
D. Hilbert and P. Bernays, Grundlagen der Mathematik, Vol. 1, Berlin, 1934.
F. Klein, Vergleichende Betrachutngen über neuere geometrische Forschungen, Verlag A. Deicher, Erlangen, 1872.
A. Robinson, A result on consistency and its application to the theory of definition, Indag. Math. 18 (1956), 47–58.
L. Svenonius, A theorem on permutations in models, Theoria (Lund) 25 (1959).
A. Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica 1, (1936).
A. Tarski, Einige Methodologische Untersuchungen über die Definierbarkeit der Begriffe Erkenntnis 5 (1935).
J.B. Wright, Quasi-projective geometry of two dimensions, Michigan Math. J. 2 (1953–4).
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© 1990 Springer-Verlag New York Inc.
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Buchi, J.R., Danhof, K.J. (1990). Definibility in Normal Theories. In: Mac Lane, S., Siefkes, D. (eds) The Collected Works of J. Richard Büchi. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8928-6_15
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DOI: https://doi.org/10.1007/978-1-4613-8928-6_15
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