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Ultraproducts and Higher Order Formulas

Which ultraproducts preserve the validity of formulas of higher order logics? To answer this question, we will introduce natural topologies (which we call ultratopologies) on ultraproducts. We will show, that ultraproducts preserving certain higher order formulas can be characterized in terms of these topologies. As an application of the above results, we provide a constructive, purely model theoretic characterization for classes definable by second order existential formulas.