The combinatorics of reasonable ultrafilters

We are interested in generalizing part of the theory of ultrafilters on omega to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal

Generic Large Cardinals and Systems of Filters

We introduce the notion of $\mathcal{C}$-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties.Comment: 36 page

Looking beyond endotoxin: a comparative study of pyrogen retention by ultrafilters used for the preparation of sterile dialyis fluid

Sterile single-use ultrafilters are used in dialysis for the preparation of the substitution fluid given to patients undergoing dialysis treatments with high convective fluid removal. The retention of pyrogenic agents by the ultrafilters is crucial to avoiding inflammatory responses. The performance of a new single-use ultrafilter (NUF) with a positively charged flat sheet membrane of relatively small membrane area and large pore size was compared to a reference ultrafilter (RUF) with a hollow fiber membrane. Filter performance was tested with various pyrogen-contaminated dialysis fluids by direct pyrogen quantification and by measuring inflammatory responses in cell-based bioassays. The NUF completely retained oligodeoxynucleotides (ODN), whereas the RUF was fully permeable. Both filters tended to decrease biological activity of DNA in filtered bacterial lysates. The NUF reduced lipopolysaccharides (LPS) and LPS-induced biological activity by 100%, whereas the RUF produced filtrates with low but detectable levels of LPS in most cases. Peptidoglycans (PGN) were fully retained both by the NUF and the RUF. The new ultrafilter retained biologically active ODN, which has not yet been described for any other device used in dialysis, and it showed better or equal retention of LPS and PGN even with a smaller membrane surface and larger pore size

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 exists and let ACA_0^{\omega} be the higher order extension of ACA_0. We show that ACA_0^{\omega}+U is \Pi^1_2-conservative over ACA_0^{\omega} and thus that ACA_0^{\omega}+\U is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly \Pi^1_2 statement \forall f \exists g A(f,g) in ACA_0^{\omega}+U a realizing term in G\"odel's system T can be extracted. This means that one can extract a term t, such that A(f,t(f))

Survey on the Tukey theory of ultrafilters

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 canonical forms for cofinal maps on ultrafilters are reviewed, as well as their applications to finding which structures embed into the Tukey types of ultrafilters. With the addition of some Ramsey theory, fine analyses of the structures at the bottom of the Tukey hierarchy are made.Comment: 25 page

On regular ultrafilters, Boolean ultrapowers, and Keisler's order

In this paper we analyse and compare two different notions of regularity for filters on complete Boolean algebras. We also announce two results from a forthcoming paper in preparation, which provide a characterization of Keisler's order in terms of Boolean ultrapowers