Matrix iterations and Cichon's diagram

Using matrix iterations of ccc posets, we prove the consistency with ZFC of some cases where the cardinals on the right hand side of Cichon's diagram take two or three arbitrary values (two regular values, the third one with uncountable cofinality). Also, mixing this with the techniques in J Symb Log 56(3):795-810, 1991, we can prove that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon's diagram.Comment: 14 pages, 2 figures, article in press for the journal Archive for Mathematical Logi

Template iterations with non-definable ccc forcing notions

We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $\kappa$ is a measurable cardinal and $\theta<\kappa<\mu<\lambda$ are uncountable regular cardinals, then there is a ccc poset forcing $\mathfrak{s}=\theta<\mathfrak{b}=\mu<\mathfrak{a}=\lambda$. Another application is to get models with large continuum where the groupwise-density number $\mathfrak{g}$ assumes an arbitrary regular value.Comment: To appear in the Annals of Pure and Applied Logic, 45 pages, 2 figure

Many Different Uniformity Numbers of Yorioka Ideals

Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise different. In addition we show that, in the same forcing extension, for two other types of simple cardinal characteristics parametrised by reals (localisation and anti-localisation cardinals), for uncountably many parameters the corresponding cardinals are pairwise different.Comment: 29 pages, 4 figure

Lebesgue measure zero modulo ideals on the natural numbers

We propose a reformulation of the ideal $\mathcal{N}$ of Lebesgue measure zero sets of reals modulo an ideal $J$ on $\omega$, which we denote by $\mathcal{N}_J$. In the same way, we reformulate the ideal $\mathcal{E}$ generated by $F_\sigma$ measure zero sets of reals modulo $J$, which we denote by $\mathcal{N}^*_J$. We show that these are $\sigma$-ideals and that $\mathcal{N}_J=\mathcal{N}$ iff $J$ has the Baire property, which in turn is equivalent to $\mathcal{N}^*_J=\mathcal{E}$. Moreover, we prove that $\mathcal{N}_J$ does not contain co-meager sets and $\mathcal{N}^*_J$ contains non-meager sets when $J$ does not have the Baire property. We also prove a deep connection between these ideals modulo $J$ and the notion of nearly coherence of filters (or ideals). We also study the cardinal characteristics associated with $\mathcal{N}_J$ and $\mathcal{N}^*_J$. We show their position with respect to Cicho\'n's diagram and prove that no further inequalities can be proved in relation with other very classical cardinal characteristics of the continuum, leaving just very few open questions. To achieve this, we discovered a new characterization of $\mathrm{add}(\mathcal{N})$ and $\mathrm{cof}(\mathcal{N})$. We also show that, in Cohen model, we can obtain many different values to the cardinal characteristics associated with our new ideals.Comment: 33 pages, 6 figure