Syntax and Semantics of the Logic $\mathcal{L}^\lambda_{\omega\omega}$

Notre Dame Journal of Formal Logic 38 (3):374-384 (1997)
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Abstract

In this paper we study the logic $\mathcal{L}^\lambda_{\omega\omega}$, which is first-order logic extended by quantification over functions . We give the syntax of the logic as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of $\mathcal{L}^\lambda_{\omega\omega}$ with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting-valued models. The logic $\mathcal{L}^\lambda_{\omega\omega}$ is the strongest for which Heyting-valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes

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References found in this work

Completeness in the theory of types.Leon Henkin - 1950 - Journal of Symbolic Logic 15 (2):81-91.
Sheaves and Logic.M. P. Fourman, D. S. Scott & C. J. Mulvey - 1983 - Journal of Symbolic Logic 48 (4):1201-1203.
La logique Des topos.André Boileau & André Joyal - 1981 - Journal of Symbolic Logic 46 (1):6-16.
Constructive Sheaf Semantics.Erik Palmgren - 1997 - Mathematical Logic Quarterly 43 (3):321-327.
Classifying toposes for first-order theories.Carsten Butz & Peter Johnstone - 1998 - Annals of Pure and Applied Logic 91 (1):33-58.

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