Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, {thus} saturating any stable theory. We then prove directly that a "bottleneck" in the inductive construction of a regular ultrafilter on $λ$ (i.e. a point after which all antichains of $P(λ)/D$ have cardinality less than $λ$) essentially prevents any subsequent ultrafilter from being flexible, {thus} from saturating any non-low theory. The paper's three main constructions are as follows. First, we construct a regular filter $D$ on $λ$ so that any ultrafilter extending $D$ fails to $λ^+$-saturate ultrapowers of the random graph, {thus} of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal $κ$, we construct a regular ultrafilter on $λ> κ$ which is $λ$-flexible but not $κ^{++}$-good, improving our previous answer to a question raised in Dow 1975. Third, assuming a weakly compact cardinal $κ$, we construct an ultrafilter to show that $\lcf(\aleph_0)$ may be small while all symmetric cuts of cofinality $κ$ are realized. Thus certain families of pre-cuts may be realized while still failing to saturate any unstable theory.
We study ultrafilters on countable sets and reaping families which are indestructible by Sacks forcing. We deal with the combinatorial characterization of such families and we prove that every reaping family of size smaller than the continuum is Sacks indestructible. We prove that complements of many definable ideals are Sacks reaping indestructible, with one notable exception, the complement of the ideal $\mathcal Z$ of sets of asymptotic density zero. We investigate the existence of Sacks indestructible ultrafilters and prove that every Sacks indestructible ultrafilter is a $\mathcal Z$-ultr…
An assortments of `species' of families of subsets of a set and some of their properties are investigated, with an eye on the logic and limit role they may play as `relaxed parallels' to ultrafilters.
We apply the concept of generic existence to p-point, q, and selective independent families that complements and emulates the ultrafilter generic existence results from Canjar and Ketonen.
We study the interplay between properties of measures on a Boolean algebra A and forcing names for ultrafilters on A. We show that several well known measure theoretic properties of Boolean algebras (such as supporting a strictly positive measure or carrying only separable measures) have quite natural characterizations in the forcing language. We show some applications of this approach. In particular, we reprove a theorem of Kunen saying that in the classical random model there are no towers of height $ω_2$.
We provide a proof that analytic almost disjoint families of infinite sets of integers cannot be maximal using a result of Bourgain about compact sets of Baire class one functions. Inspired by this and related ideas, we then provide a new proof of that there are no maximal almost disjoint families in Solovay's model. We then use the ideas behind this proof to provide an extension of a dichotomy result by Rosenthal and by Bourgain, Fremlin and Talagrand to general pointwise bounded functions in Solovay's model. We then show that the same conclusions can be drawn about the model obtained when we…
Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal $\mathfrak r$. This is achieved through analyzing nowhere reaping families of subsets of $\mathbb N$ and through applying a paving lemma which is a consequence of a p…
We consider compactness characterizations of large cardinals. Based on results of Benda \cite{b-sccomp}, we study compactness for omitting types in various logics. In $\bL_{κ, κ}$, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic. In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vopěnka's Principle.
We introduce the property ``$F$-linked'' of subsets of posets for a given free filter $F$ on the natural numbers, and define the properties ``$μ$-$F$-linked'' and ``$θ$-$F$-Knaster'' for posets in a natural way. We show that $θ$-$F$-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct $θ$-$\mathrm{Fr}$-Knaster posets (where $\mathrm{Fr}$ is the Frechet ideal) via matrix iterations of ${<}θ$-ultrafilter-linked posets (restricted to some level of the matrix). This is a…
我们提出了一个新的，博弈论，方法的理想化的强迫，在融合游戏。这推广了经典的方法，以袋和米勒强迫。对于可定义的（$\mathbfΠ^1_1$ on $\mathbfΣ^1_1) $ σ $-ideals we show that if a $ σ $-ideal is generated by closed sets, then it is generated by closed sets in all forcing extensions. We also prove an infinite-dimensional version of the Solecki dichotomy for analytic sets. Among examples, we investigate the $ σ $-ideal $ \E $ generated by closed null sets and $ σ$-理想与非分段连续函数相连.
