On The Model Of Hyperrational Numbers With Selective Ultrafilter

Quasi-selective and weakly Ramsey ultrafilters

Selective ultrafilters are characterized by many equivalent properties, in particular the Ramsey property that every finite colouring of unordered pairs of integers has a homogeneous set in U, and the equivalent property that every function is nondecreasing on some set in U. Natural weakenings of these properties led to the inequivalent notions of weakly Ramsey and of quasi-selective ultrafilter, introduced and studied in [1] and [4], respectively. U is weakly Ramsey if for every finite colouring of unordered pairs of integers there is a set in U whose pairs share only two colours, while U is …

Quasi-selective ultrafilters and asymptotic numerosities

We isolate a new class of ultrafilters on N, called "quasi-selective" because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of quasi-selective ultrafilters is equivalent to the existence of "asymptotic numerosities" for all sets of tuples of natural numbers. Such numerosities are hypernatural numbers that generalize finite cardinalities to countable point sets. Most notably, they maintain the structure of ordered semiring, and, in a precise sense, they allow for a natural extension of asymptot…

A notion of selective ultrafilter corresponding to topological Ramsey spaces

We introduce the relation of "almost-reduction" in an arbitrary topological Ramsey space R, as a generalization of the relation of "almost-inclusion" on the space of infinite sets of natural numbers (the Ellentuck space). This leads us to a type of ultrafilter U on the set of first approximations of the elements of R which corresponds to the well-known notion of "selective ultrafilter" on N, the set of natural numbers. The relationship turns out to be rather exact in the sense that it permits us to lift several well-known facts about selective ultrafilters on N and the Ellentuck space to the u…

向Shelah的无无处稠密超滤子模型添加超滤子

Adding ultrafilters to Shelah's model for no nowhere dense ultrafilters

本文通过一种满足完全萨克斯性质的力迫方法，构造了包含$(2, \aleph_0)$-选择性超滤子但无无处稠密超滤子的模型，证明前者存在性不蕴含后者。

The next best thing to a P-point

We study ultrafilters on $ω^2$ produced by forcing with the quotient of $\scr P(ω^2)$ by the Fubini square of the Fréchet filter on $ω$. We show that such an ultrafilter is a weak P-point but not a P-point and that the only non-principal ultrafilters strictly below it in the Rudin-Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at …

稳定序并集与选择性超滤子

Stable ordered-union versus selective ultrafilters

本文证明了存在至少两个非同构的选择性超滤子且不存在稳定序并集超滤子是相容的，回答了 Blass 于 1987 年提出的相关问题。