Ultraproduct invariant logics

Abstract

Abstract. In this paper we give a construction of logics via the property of being preserved from the models to their ultraproduct. Specific examples are given which include some cardinality quantifiers.We will assume that the reader is familiar with basic model theory including ultraproducts, Chang and Keisler Take Mod to be a class of generalized models of the same type closed under restriction and expansion of languages.Let SeqK(Mod) be the class of all sequences of length « of elements of Mod, Ult(«) be the set of ultrafilters on «, and Seq(Mod) = UK SeqK(Mod) and Ult= UKUlt(«).Definition.An ultraproduct on Mod is a partial function ULT from U, SeqK(Mod) X Ult(«) into Mod.We say that the pair <Mod, ULT) is closed under «-ultraproducts if Seq^Mod) X Ult(ic) G Dom ULT. If <Mod, ULT> is closed under «-ultraproducts for all cardinals « then it is said to be closed under ultraproducts. Let <E2, Ne2> be the second-order logic with additional quantification over the generalized domains. That is: Let 7(Mod, ULT) be the sublogic of £2 consisting of the invariant sentences. Let Ult*(«) be the class of regular ultrafilters over «.