arXiv subjects
Publications and source records attributed to Jonathan Cancino-Manríquez.
We present three models concerning Tukey types of ultrafilters on $ω$. The first model is built via a countable support iteration, and we show there is no basically generated ultrafilter in such model. The second and third models are built upon different and novel techniques, and in such models all ultrafilters are Tukey top, thus providing an answer to the Isbell problem. In all models there is no $\mathsf{nwd}$-ultrafilter.
We answer in negative the problem if the existence of a P-measure implies the existence of a P-point. Namely, we show that if we add random reals to a certain unique P-point model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by $ω$ product of Silver forcings and that in the model obtained by the product of $ω_2$ many Silver forcings there are no P-measures of countable Maharam type
A family $\mathcal{A} \subseteq [ω]^ω$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal A$ and $A \in \mathcal{A} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i \in n} X_i$ is infinite, is said to be ideal independent. We prove that an ideal independent family $\mathcal{A}$ is maximal if and only if $\mathcal A$ is $\mathcal J$-completely separable and maximal $\mathcal J$-almost disjoint for a particular ideal $\mathcal J$ on $ω$. We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, where $\mathfrak{s}_{mm}$ is the minimal cardinality of maximal ideal independent family. This, in particular, establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality $λ$ for each $λ\in C$, thus establishing the consistency of $C\subseteq \hbox{spec}(\mathfrak{s}_{mm})$. Assuming $\mathsf{CH}$, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, $^ωω$-bounding, $p$-point preserving forcing notion and evaluate $\mathfrak{s}_{mm}$ in several well studied forcing extensions. We also study natural filters associated with ideal independence and introduce an analog of Mrówka spaces for ideal independent families.
A family $\mathscr{I} \subseteq [ω]^ω$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal I$ and $A \in \mathscr{I} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i < n} X_i$ is infinite, is said to be ideal independent. An ideal independent family which is maximal under inclusion is said to be a maximal ideal independent family and the least cardinality of such family is denoted $\mathfrak{s}_{mm}$. We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, which in particular establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality $λ$ for each $λ\in C$, thus establishing the consistency of $C\subseteq \hbox{spec}(\mathfrak{s}_{mm})$. Assuming $\mathsf{CH}$, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, $^ωω$-bounding, $p$-point preserving forcing notion and evaluate $\mathfrak{s}_{mm}$ in several well studied forcing extensions.