The Skolem-löwenheim theorem in toposes

Studia Logica 42 (4):461 - 475 (1983)
The topos theory gives tools for unified proofs of theorems for model theory for various semantics and logics. We introduce the notion of power and the notion of generalized quantifier in topos and we formulate sufficient condition for such quantifiers in order that they fulfil downward Skolem-Löwenheim theorem when added to the language. In the next paper, in print, we will show that this sufficient condition is fulfilled in a vast class of Grothendieck toposes for the general and the existential quantifiers.
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    George Loullis (1979). Sheaves and Boolean Valued Model Theory. Journal of Symbolic Logic 44 (2):153-183.
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