Let (M,X ) |= ACA0 be such that PX , the collection of all unbounded sets in X , admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in X such that M thinks T is consistent. We prove that there is an end-extension N |= T of M such that the subsets of M coded in N are precisely those in X . As a special case we get that any Scott set with a defina...

We introduce the concept of the generic existence of P-point, Qpoint, and selective ultrafilters, a concept which is somewhat stronger than the existence of these sorts of ultrafilters. We show that selective ultrafilters exist generically iff semiselectives do iff mc = c, and we show that ß-point ultrafilters exist generically iff semiß-points do iff mc — d , where d is the minimal cardinality...

We prove that every continuum of weight א1 is a continuous image of the Čech-Stone-remainder R of the real line. It follows that under CH the remainder of the half line [0,∞) is universal among the continua of weight c. We complement this result by showing that 1) under MA every continuum of weight less than c is a continuous image of R, 2) in the Cohen model the long segment of length ω2 + 1 i...

Reduced products are generalizations of ultraproducts where the filter used need not be an ultrafilter. With a suitable choice of filter, we can then get a more constructive model of non-standard analysis. We study properties of such reduced products and investigate what classical results are still valid in a constructive setting. A boundedness principle BD, not derivable in pure constructive m...

We prove that o(κ) = κ is sufficient to construct a model V [C] in which κ is measurable and C is a closed and unbounded subset of κ containing only inaccessible cardinals of V . Gitik proved that o(κ) = κ is necessary. We also calculate the consistency strength of the existence of such a set C together with the assumption that κ is Mahlo, weakly compact, or Ramsey. In addition we consider the ...

This article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell’s Problem and the question of how closely related the Rudin-Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guarantee that an ultrafilter is below the Tukey maximum. The known ca...

We present some explicit constructions of universal R-trees with applications to the asymptotic geometry of hyperbolic spaces. In particular, we show that any asymptotic cone of a complete simply connected manifold of negative curvature is a complete homogeneous R-tree with the valency 20 at every point. It implies that all these asymptotic cones are isometric depending neither on a manifold no...

In a seminal contribution, Hansson (1976) demonstrates that the collection of decisive coalitions associated with an Arrovian social welfare function forms an ultrafilter. He goes on to show that if transitivity is weakened to quasi-transitivity as the coherence property imposed on a social relation, the set of decisive coalitions is a filter. We examine the notion of decisiveness with acyclica...

Łoś’s theorem for (bounded) D-ultrapowers, D being the ultrafilter introduced by Kanovei and Shelah [Journal of Symbolic Logic, 69(1):159–164, 2004], can be established within Zermelo–Fraenkel set theory plus Countable Choice (ZF+ACω). Thus, the Transfer Principle for both Kanovei and Shelah’s definable nonstandard model of the reals and Herzberg’s definable nonstandard enlargement of the super...

2014 We use a set theoretic approach to consensus by viewing an object as a set of smaller pieces called "bricks". A consensus function is neutral if there exists a family D of sets such that a brick s is in the output of a profile if and only if the set of positions with objects that contain s belongs to D. We give sufficient set theoretic conditions for D to be a lattice filter and, in the ca...