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Godel Logics and descriptive properties of the reals


This project aims at a thorough mathematical connection of descriptive properties of the reals with semantic properties of Godel logics. If we want to discuss logical descriptions of descriptive, i.e. topological and order theoretic, properties, an adequate logic has to be determined by the order structure of the underlying truth value set, and should not depend on quantitative properties. With other words, validity should be invariant with respect to adequate classes of non-trivial isomorphism. The most relevant many-valued logics of this kind are the Godel logics. We will determine to which extent the language of Godel logics is able to represent structural properties of the reals, thus characterizing the expressive power of Godel logics. On the other hand we want to find properties of the reals which provide suitable classifications, i.e. intentional definitions of Godel logics.The expressive power of a logic, i.e. the range of properties representable within the logic, is of great importance as soon as these logics should be put to use. It defines the limits of what can be achieved and what problems can be formulated. For some formalizations of Godel logics (propositional, quantified propositional and w.r.t. propositional entailment) this question has been settled, but for the most important formalization, the first-order Godel logics, results are still scarce, but suggest that an extension of the Cantor-Bendixson Theorem may serve as main method to obtain a characterization of expressiveness. (The original variant of the Cantor-Bendixson Theorem has been successfully applied by the applicant to characterize the recursive enumerability of first-order Godel logics.) An extended Cantor-Bendixson algorithm, which connects the topological analysis with order properties, will also allow providing a complete classification of Godel logics, i.e. as we try to find a characterization of Godel logics in the language of descriptive properties of the reals.

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Banchi di sotto, 55

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