Cascades, order, and ultrafilters

Independence, order, and the interaction of ultrafilters and theories

We consider the question, of longstanding interest, of realizing types in regular ultrapowers. In particular, this is a question about the interaction of ultrafilters and theories, which is both coarse and subtle. By our prior work it suffices to consider types given by instances of a single formula. In this article, we analyze a class of formulas φ whose associated characteristic sequence of h...

Non-principal Ultrafilters, Program Extraction and Higher Order Reverse Mathematics

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher order arithmetic. Let (U) be the statement that a non-principal ultrafilter on N exists and let ACA0 be the higher order extension of ACA0. We show that ACA0 + (U) is Π2-conservative over ACA0 and thus that ACA0 +(U) is conservative over PA. Moreover, we provide a program extraction method and s...

A Remark on the Rudin-keisler Order of Ultrafilters

Define a (pre)order on /3co as follows: p • q iff there is a finite to one function f c woo such that/3f(cl) = p. Let ,•{ denote he Rudin-Keisler order on Ow. We show that here are points p,qCw* such that Px•q, but p and q are •<-incomparable. 0. Introduction. In [5], M. E. Rudin showed that if there are points p,q G co* Such that for any finite to one function f G cow we have that 13f(p) 4:13f...

Neighborhoods, Ultrafilters, and Canonicity

This paper will remind the reader of neighborhood semantics for modal logics, compare them with relational semantics, and then look at some questions about neighborhood semantics that have been answered, and some which are still open. We will then introduce “ultrafilter semantics,” a way of expressing all sets over a canonical frame in an ‘effable’ way. This provides us with a conceptually easy...

Ultrafilters and Abelian Groups