We prove that the moduli of U-convexity, introduced by Gao (1995), of the ultrapower X̃ of a Banach space X and of X itself coincide whenever X is super-reflexive. As a consequence, some known results have been proved and improved. More precisely, we prove that uX(1) > 0 implies that both X and the dual space X∗ of X have uniform normal structure and hence the “worth” property in Corollary 7 of ...

The ultrapower T * of an arbitrary ordered set is introduced as infinitesimal extension . It obtained the equivalence classes sequences in , where corresponding relation generated by a free ultrafilter on natural numbers. established that always satisfies Cantor’s property, while one can give necessary and sufficient conditions for so would be complete or it fulfill open completeness respective...

This paper is a contribution to the study of equality-free logic, that is, first-order logic without equality. We mainly devote ourselves to the study of algebraic characterizations of its relation of elementary equivalence by providing some Keisler-Shelah type ultrapower theorems and an Ehrenfeucht-Fraı̈ssé type theorem. We also give characterizations of elementary classes in equalityfree logic...

It is consistent (relative to ZFC) that each union of max{b, g} many families in the Baire space ω which are not finitely dominating is not dominating. In particular, it is consistent that for each nonprincipal ultrafilter U , the cofinality of the reduced ultrapower ω/U is greater than max{b, g}. The model is constructed by oracle chain condition forcing, to which we give a self-contained intr...

Let T1, T2 be countable first-order theories, Mi |= Ti, and D any regular ultrafilter on λ ≥ א0. A longstanding open problem of Keisler asks when T2 is more complex than T1, as measured by the fact that for any such λ,D, if the ultrapower (M2)/D realizes all types over sets of size ≤ λ, then so must the ultrapower (M1)/D. In this paper, building on the author’s prior work [11] [12] [13], we sho...

Say that an elementary embedding j : N → M is cardinal preserving if CAR = CAR = CAR. We show that if PFA holds then there are no cardinal preserving elementary embeddings j : M → V . We also show that no ultrapower embedding j : V → M induced by a set extender is cardinal preserving, and present some results on the large cardinal strength of the assumption that there is a cardinal preserving j...

This application of nonstandard analysis utilizes the notion of the highly-saturated enlargement. These nonstandard methods are applied to the theory of generalized functions (distributions) and demonstrates how such analysis clarifies many aspects of this theory. 1. Additional Modeling Concepts. In what follows, the basic notation and definitions are as they appear in [3]. However, except as m...

The relative commutant A∩A of a strongly self-absorbing algebra A is indistinguishable from its ultrapower A . This applies both to the case when A is the hyperfinite II1 factor and to the case when it is a strongly self-absorbing C-algebra. In the latter case we prove analogous results for l∞(A)/c0(A) and reduced powers corresponding to other filters on N. Examples of algebras with approximate...

We prove that a Spector–like ultrapower extension N of a countable Solovay model M (where all sets of reals are Lebesgue measurable) is equal to the set of all sets constructible from reals in a generic extension M[α] where α is a random real over M. The proof involves an almost everywhere uniformization theorem in the Solovay model. Research supported by the Netherlands Organization for Scient...

A long-standing question in the theory of measures noncompactness is that for Kuratowski measure α defined on a metric space M, and every bounded subset B⊂M, there countable B0⊂B such α(B0)=α(B)? In this paper, we give an affirmative answer to above. It done by showing each nonempty set B Banach space, so strongly finitely representable B0, free ultrafilter U affinely isometric ultrapower [co(B...