In the ten years since the celebration of Professor Banaschewski’s 60th birthday (and the writing of [4] in commemoration), there has been a fair amount of development in the theory of ultracoproducts of compacta (i.e., compact Hausdorff spaces). Papers [3, 5, 7, 6] have been written by this author; also there is the paper [13] by the late R. Gurevič. In addition to this, there has been a paral...

The Levy-Solovay Theorem [LevSol67] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong card...

This paper contributes to the set-theoretic side of understanding Keisler’s order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality lcf(א0,D) of א0 modulo D, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as se...

Let D be an ultrafilter on the set N of natural numbers. To each function p: N — N and each ultrafilter E that is mapped to D by p, we associate a Dedekind cut in the ultrapower ö-prod N. We characterize, in terms of rather simple closure conditions, the cuts obtainable in this manner when various restrictions are imposed on E and p. These results imply existence theorems, some known and some n...

We give two characterizations of conservative extensions of models of arithmetic, in terms of the existence and uniqueness of certain amalg~mations with other models. We also establish a connection between conservativity and some combinatorial properties of ultrafilter mappings. Arithmetic is, in this paper, the complete theory of the structure N whose universe is the set of natural numbers and...

The findings reported in this paper aim to garner the interest of both model theorists and operator algebraists alike. Using a novel blend theoretic algebraic methods, we show that family II 1 factors elementarily equivalent hyperfinite factor R all admit embeddings into U with factorial relative commutant. This answers long standing question Popa for an uncountable factors. We introduce notion...

Abstract Given a bounded convex subset C of Banach space X and free ultrafilter $${\mathcal {U}}$$ U , we study which points $$(x_i)_{\mathcal ( x i ) are extreme the ultrapower $$C_{\mathcal <mml...

We prove the statement in the title.

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there ...