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      Interpolation in extensions of first-order logic

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          Abstract

          We provide a constructive proof of the interpolation theorem for extensions of classical first order logic with a special type of geometric axioms, called singular geometric axioms. As a corollary, we obtain a direct proof of interpolation for first-order logic with identity.

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          Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory

          One task of metamathematics is to relate suggestive but nonelementary modeltheoretic concepts to more elementary proof-theoretic concepts, thereby opening up modeltheoretic problems to proof-theoretic methods of attack. Herbrand's Theorem (see [8] or also [9], vol. 2) or Gentzen's Extended Hauptsatz (see [5] or also [10]) was first used along these lines by Beth [1]. Using a modified version he showed that for all first-order systems a certain modeltheoretic notion of definability coincides with a certain proof theoretic notion. In the present paper the Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms. This may be interpreted as showing that (apart from some relatively minor exceptions which will be made apparent below) the expressive power of each first-order system is rounded out, or the system is functionally complete , in the following sense: Any functional relationship which obtains between concepts that are expressible in the system is itself expressible and provable in the system. A second application is concerned with the hierarchy of second-order formulas. A certain relationship is shown to hold between first-order formulas and those second-order formulas which are of the form (∃T 1 )…(∃T k )A or (T 1 )…(T k )A with A being a first-order formula. Modeltheoretically this can be regarded as a relationship between the class AC and the class PC ⊿ of sets of models, investigated by Tarski in [12] and [13].
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            3. Investigations into Logical Deduction

            (1969)
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              Cut Elimination in the Presence of Axioms

              A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic with equality in which also cuts on the equality axioms are eliminated.
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                Author and article information

                Journal
                31 July 2018
                Article
                1807.11848
                3ee1c905-6299-4f1d-b0cf-b0bdfe0789e7

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                math.LO

                Logic & Foundation
                Logic & Foundation

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