Associated to each ultrafilter U on ω and each map p : ω → ω is a Dedekind cut in the ultrapower ωω/p(U). Blass has characterized, under CH, the cuts obtainable when U is taken to be either a p-point ultrafilter, a weakly-Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorcevic have introduced the topological Ramsey space R1. Associated to the space R1 is a notion of Ramsey ultrafilt...

Associated to each ultrafilter U on ω and each map p : ω → ω is a Dedekind cut in the ultrapower ω/p(U). Blass has characterized, under CH, the cuts obtainable when U is taken to be either a p-point ultrafilter, a weakly-Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorcevic have introduced the topological Ramsey spaceR1. Associated to the spaceR1 is a notion of Ramsey ultrafilter ...

We study ultraproducts of finite residue rings $\underset{n\in\mathbb{N}}{\prod}\mathbb{Z}/n\mathbb{Z} \diagup \mathcal{U} $ where $\mathcal{U}$ is a non-principal ultrafilter. find sufficient conditions the ultrafilter to determine if resulting ultraproduct \mathcal{U}$ has simple, NIP, NTP2 but not simple nor or TP2 theory, noting that all these four cases occur.

There exist two known types of ultrafilter extensions first-order models, both in a certain sense canonical. One them (Goranko Filter and structures: universal-algebraic aspects, preprint, 2007) comes from modal logic universal algebra, fact goes back to Jónsson Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev Lect Notes Comput Sci 6521:162–177, 2011; Saveliev ...

We study the definability of ultrafilter bases on $$\omega $$ in sense descriptive set theory. As a main result we show that there is no coanalytic base for Ramsey ultrafilter, while L can construct $$\Pi ^1_1$$ P-point and Q-point bases. also existence $${\varvec{\Delta }}^1_{n+1}$$ equivalent to $${\varvec{\Pi }}^1_n$$ base, $$n \in \omega . Moreover introduce Borel version classical number m...

2 Filter: Filters and Ultrafilters 13 2.1 Definitions and basic properties . . . . . . . . . . . . . . . . . 13 2.1.1 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 Free Ultrafilters . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Collect properties . . . . . . . . . . . . . . . . . . . . . . . . . ...

It is well known that, in a topological space, the open sets can be characterized using filter convergence. In ZF (Zermelo-Fraenkel set theory without the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the ultrafilter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ult...

A method is described by which poliovirus can be rapidly and simply concentrated by the use of a Diaflo XM-50 ultrafilter membrane. Freon-extracted ultrafilter concentrates banded in CsCl provided a 1,724-fold volumetric concentration of poliovirus. During concentration, trypsin-digested cellular material can pass through the ultrafilter membrane, thus providing a versatile means of degrading a...