Diego Alejandro Mejia

Shizuoka University · Creative Science Course (Mathematics)

Based on the work of Shelah, Kellner, and Tȃnasie (Fund. Math., 166(1-2):109-136, 2000 and Comment. Math. Univ. Carolin., 60(1):61-95, 2019), and the recent developments in the third author's master's thesis, we develop a general theory of iterated forcing using finitely additive measures. For this purpose, we introduce two new notions: on the one...

Denote by NA and MA the ideals of null-additive and meager-additive subsets of 2^ω , respectively. We prove in ZFC that add(NA) = non(NA) and introduce a new (Polish) relational system to reformulate Bartoszyński's and Judah's characterization of the uniformity of MA, which is helpful to understand the combinatorics of MA and to prove consistency r...

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 $\...

The notion of θ-FAM-linkedness, introduced in the second author's master thesis, is a formalization of the notion of strong FAM limits for intervals, whose initial form and applications have appeared in the work of Saharon Shelah, Jakob Kellner, and Anda Tȃnasie, for controlling cardinals characteristics of the continuum in ccc forcing extensions....

Let $\mathcal{SN}$ be the $\sigma$-ideal of the strong measure zero sets of reals. We present general properties of forcing notions that allow to control of the additivity of $\mathcal{SN}$ after finite support iterations. This is applied to force that the four cardinal characteristics associated with $\mathcal{SN}$ are pairwise different: \[\mathr...

This paper is intended to survey the basics of localization and anti-localization cardinals on the reals, and its interplay with notions and cardinal characteristics related to measure and category.

We improve the previous work of Yorioka and the first author about the combinatorics of the ideal $\mathcal{SN}$ of strong measure zero sets of reals. We refine the notions of dominating systems of the first author and introduce the new combinatorial principle $\mathrm{DS}(\delta)$ that helps to find simple conditions to deduce $\mathfrak{d}_\kappa...

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 $\sigma$...

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal \({{\mathcal {S}}}{{\mathcal {N}}}\). As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical car...

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 (localisa...

We use known finite support iteration techniques to present various examples of models where several cardinal characteristics of Cicho\'n's diagram are pairwise different. We show some simple examples forcing the left-hand side of Cicho\'n's diagram, and present the technique of restriction to models to force Cicho\'n's maximum (original from Golds...

Let q be a prime. We classify the odd primes p ≠ q such that the equation x2 ≡ q (mod p) has a solution, concretely, we find a subgroup L4q of the multiplicative group U4q of integers relatively prime with 4q (modulo 4q) such that x2 ≡ q (mod p) has a solution iff p ≡ c (mod 4q) for some c ∈ L4q. Moreover, L4q is the only subgroup of U4q of half or...

Let $q$ be a prime. We classify the odd primes $p\neq q$ such that the equation $x^2\equiv q\pmod{p}$ has a solution, concretely, we find a subgroup $\mathbb{L}_{4q}$ of the multiplicative group $\mathbb{U}_{4q}$ of integers relatively prime with $4q$ (modulo $4q$) such that $x^2\equiv q\pmod{p}$ has a solution iff $p\equiv c\pmod{4q}$ for some $c\...

We show how to construct, via forcing, splitting families that are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń’s diagram, $$\mathfrak{m}$$ m (2-Knaster), $$\mathfrak{p...

Combining creature forcing approaches from arXiv:1003.3425 and arXiv:1402.0367, we show that, under CH, there is a proper $\omega^\omega$-bounding poset with $\aleph_2$-cc that forces continuum many pairwise different cardinal characteristics, parametrised by reals, for each one of the following six types: uniformity and covering numbers of Yorioka...

We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new [Formula: see text]-sequences (for some regular [Formula: see text]). As an application, we show that consistently the following cardinal characteristics can be different: The (“independent”) characteristics in Cichoń’s diagram, plus [Formula:...

We show how to construct, via forcing, splitting families than are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cicho\'n's diagram, $\mathfrak{m}(2\text{-Knaster})$, $\mathfra...

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, w...

We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new ${<}\kappa$-sequences (for some regular $\kappa$). As an application, we show that consistently the following cardinal characteristics can be different: The ("independent") characteristics in Cicho\'n's diagram, plus $\aleph_1<\mathfrak m<\math...

By coding Polish metric spaces with metrics on countable sets, we propose an interpretation of Polish metric spaces in models of ZFC and extend Mostowski's classical theorem of absoluteness of analytic sets for any Polish metric space in general. In addition, we prove a general version of Shoenfield's absoluteness theorem.

Cicho\'n's diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and $\sigma$-compact subsets of the irrationals. It is consistent that all entries of Cicho\'n's diagram are pairwise different (apart from $\textrm{add}(\mathcal{M})$ and $\text...

We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new ${<}\kappa$-sequences (for some regular $\kappa$). As an application, we present models (assuming the consistency of three strongly compact cardinals) with 13 simultaneously different cardinal characteristics (including $\aleph_1$ and continuum...

In this note, we relax the hypothesis of the main results in Kellner-Shelah-T\v{a}nasie's "Another ordering of the ten cardinal characteristics in Cicho\'n's diagram".

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal $\mathcal{SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the...

We introduce the property "$F$-linked" of subsets of posets for a given free filter $F$ on the natural numbers, and define the properties "$\mu$-$F$-linked" and "$\theta$-$F$-Knaster" for posets in the natural way. We show that $\theta$-$F$-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concernin...

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 (localisa...

We use coherent systems of FS iterations on a power set, which can be seen as matrix iteration that allows restriction on arbitrary subsets of the vertical component, to prove general theorems about preservation of certain type of unbounded families on definable structures and of certain mad families (like those added by Hechler's poset for adding...

In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm Ext}^1_{KQ}(\mathcal{X},KQ)=0$ then $\mathcal{X}$ is projective. In contrast, we show that if $Q$ is a specific quiver of the type above, then ther...

We construct models, by three-dimensional arrays of ccc posets, where many classical cardinal characteristics of the continuum are pairwise different.

Yorioka introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in . We construct a matrix iteration of c.c.c. posets to force that, for many ideals in that class, their associated card...

We use countable metric spaces to code Polish metric spaces and evaluate the complexity of some statements about these codes and of some relations that can be determined by the codes. Also, we propose a coding for continuous functions between Polish metric spaces.

We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cicho\'n's diagram, in particular, a model where the diagram can be separated into 7 different values. Fur...

We prove the consistency of $\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{p}=\mathfrak{s}<\mathrm{add}(\mathcal{M})=\mathrm{cof}(\mathcal{M})<\mathfrak{a}=\mathrm{non}(\mathcal{N})=\mathfrak{c}$ with ZFC where each of these cardinal invariants assume arbitrary uncountable regular values.

Using a finite support iteration of ccc forcings, we construct a model of $\aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})=\mathfrak{c}$.

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}=\la...

We prove that, for a suitable iteration along a template, we can compute any name for a real from a Borel function coded in the ground model evaluated at only countably many of the generic reals.

The~\emph{Rothberger number} $\mathfrak{b} (\mathcal{I})$ of a definable ideal $\mathcal{I}$ on $\omega$ is the least cardinal $\kappa$ such that there exists a Rothberger gap of type $(\omega,\kappa)$ in the quotient algebra $\mathcal{P} (\omega) / \mathcal{I}$. We investigate $\mathfrak{b} (\mathcal{I})$ for a subclass of the $F_\sigma$ ideals, t...

We extend the applications of the techniques used in Arch Math Logic 52:261-278, 2013, to present various examples of consistency results where some cardinal invariants of the continuum take arbitrary regular values with the size of the continuum being bigger than $\aleph_2$.

We present the classical theory of preservation of $\sqsubset$-unbounded families in generic extensions by ccc posets, where $\sqsubset$ is a definable relation of certain type on spaces of real numbers, typically associated with some classical cardinal invariant. We also prove that, under some conditions, these preservation properties can be prese...

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. Brendle, J. Symb. Log. 56, No. 3, 795–810 (1991; Zbl 07...

Resumen. Con base en la caracterización de los números naturales a partir de la propiedad de recursión (ver [2]), probamos en forma general que para un conjunto dado las propiedades de recursión, inducción y buena fundación son equivalentes entre sí. El resultado lo extendemos a clases y lo utilizamos para dar otra prueba de la caracterización del...

Resumen. Con base en la caracterización de los números naturales a partir de la propiedad de recursión (ver [2]), probamos en forma general que para un conjunto dado las propiedades de recursión, inducción y buena fundación son equivalentes entre sí. El resultado lo extendemos a clases y lo utilizamos para dar otra prueba de la caracterización del...