Based on the well-known cardinal characteristics $\mathfrak{s}$, $\mathfrak{r}$ and $\mathfrak{i}$, we introduce nine related cardinal characteristics by using the notion of asymptotic density to characterise different intersection properties of infinite sets. We prove several bounds and consistency results, e. g. the consistency of $\mathfrak{s} < \mathfrak{s}_{1/2}$, as well as several results about possible values of $\mathfrak{i}_{1/2}$.
We prove that a variety of generalized cardinal characteristics, including meeting numbers, the reaping number, and the dominating number, satisfy an analogue of the Galvin-Hajnal theorem, and hence also of Silver's theorem, at singular cardinals of uncountable cofinality.
For a free ultrafilter U on omega we study several cardinal characteristics which describe part of the combinatorial structure of U. We provide various consistency results; e.g. we show how to force simultaneously many characters and many pi --characters. We also investigate two ideals on the Baire space omega^omega naturally related to U and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.
We prove the consistency of the statement $\mathfrak{u}_{\aleph_ω}<2^{\aleph_ω}$. We show that the consistency strength of this statement is exactly a measurable cardinal $μ$ so that $o(μ)=μ^{++}$.
Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $λ>κ$ to $κ$) such that the composition still forces the previous values to these characteristics. We also show how to force distinct values to $\mathfrak m$, $\mathfrak p$ and $\mathfrak h$ and also keeping all the values in Cichoń's diagram distint, using the Boolean Ultrapower method of arXiv:1708.03691 . (In arXiv:2006.09826 , the same was done for the newer Cichoń's Maximum construction, which avoids large cardinals…
In this note, we relax the hypothesis of the main results in Kellner-Shelah-Tǎnasie's "Another ordering of the ten cardinal characteristics in Cichoń's diagram".
Let $\mathcal{E}$ be the $σ$-ideal generated by the closed measure zero sets of reals. We use an ultrafilter-extendable matrix iteration of ccc posets to force that, for $\mathcal{E}$, their associated cardinal characteristics (i.e.\ additivity, covering, uniformity and cofinality) are pairwise different.
Much recent work in cardinal characteristics has focused on generalizing results about $ω$ to uncountable cardinals by studying analogues of classical cardinal characteristics on the generalized Baire and Cantor spaces $κ^κ$ and $2^κ$. In this note I look at generalizations to other function spaces, focusing particularly on the space of functions $f:ω^ω\to ω^ω$. By considering classical cardinal invariants on Baire space in this setting I derive a number of "higher dimensional" analogues of such cardinals, ultimately introducing 18 new cardinal invariants, alongside a framework that allows for…
Let P be the direct product of countably many copies of the additive group Z of integers. We study, from a set-theoretic point of view, those subgroups of P for which all homomorphisms to Z annihilate all but finitely many of the standard unit vectors. Specifically, we relate the smallest possible size of such a subgroup to several of the standard cardinal characteristics of the continuum. We also study some related properties and cardinals, both group-theoretic and set-theoretic. One of the set-theoretic properties and the associated cardinal are combinatorially natural, independently of any …
We provide a model where u(κ) < 2^κ for a supercompact cardinal κ. Garti and Shelah have provided a sketch of how to obtain such a model by modifying the construction in a paper of Dzamonja and Shelah; we provide here a complete proof using a different modification and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.
Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number $\mathfrak b$. We analyze some of the basic behavior of these properties, showing for instance that the property corresponding to the splitting …
We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new ${<}κ$-sequences (for some regular $κ$). As an application, we show that consistently the following cardinal characteristics can be different: The ("independent") characteristics in Cichoń's diagram, plus $\aleph_1<\mathfrak m<\mathfrak p<\mathfrak h<\mathrm{add}(\mathcal{N})$. (So we get thirteen different values, including $\aleph_1$ and continuum). We also give constructions to alternatively separate other MA-numbers (instead of $\mathfrak m$), namely: MA for $k$-Knaster from MA for $k+…
We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. While this analogy was first studied explicitly by Rupprecht in his PhD thesis, many prior results can be viewed from this perspective. After a comprehensive survey of the analogy for characteristics from Cichon's diagram, we extend it to Kurtz randomness and the analogue of the Specker-Eda number.
We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (boldface Sigma^0_2 and, under suitable restrictions, Pi^0_2) are shown to have pleasant properties, related to Baire category. We construct models of set theory where (unrestricted) boldface Pi^0_2-characteristics behave quite chaotically and no new characteristics appear at higher complexity levels. We also discuss some characteristics associated with partition theorems and we present, in an appendix, a simplified p…
Let $\mathcal{SN}$ be the strong measure zero $σ$-ideal. We prove a result providing bounds for $\mathrm{cof}(\mathcal{SN})$ which implies Yorioka's characterization of the cofinality of the strong measure zero. In addition, we use forcing matrix iterations to construct a model of ZFC that satisfies $\mathrm{add}(\mathcal{SN})=\mathrm{cov}(\mathcal{SN})<\mathrm{non}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$.
An interval algebra is a Boolean algebra which is isomorphic to the algebra of finite unions of half-open intervals, of a linearly ordered set. An interval algebra is hereditary if every subalgebra is an interval algebra. We answer a question of M. Bekkali and S. Todorčević, by showing that it is consistent that every $σ$-centered interval algebra of size $\mathfrak{b}$ is hereditary. We also show that there is, in ZFC, an hereditary interval algebra of cardinality $\aleph_1.$
We discuss two general aspects of the theory of cardinal characteristics of the continuum, especially of proofs of inequalities between such characteristics. The first aspect is to express the essential content of these proofs in a way that makes sense even in models where the inequalities hold trivially (e.g., because the continuum hypothesis holds). For this purpose, we use a Borel version of Vojtas's theory of generalized Galois-Tukey connections. The second aspect is to analyze a sequential structure often found in proofs of inequalities relating one characteristic to the minimum (or maxim…
We show: There are pairs of universes V_1 subseteq V_2 and there is a notion of forcing P in V_1 such that the change mentioned in the title occurs when going from V_1[G] to V_2[G] for a P-generic filter G over V_2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.
We continue our investigation =of Shelah's interpretability orders $\trianglelefteq^*_κ$ as well as the new orders $\trianglelefteq^\times_κ$. In particular, we give streamlined proofs of the existence of minimal unstable, unsimple and nonlow theories in these orders, and we give a similar analysis of the hypergraph examples $T_{n, k}$ of Hrushovski. We also prove that if $\mathcal{B}$ is a complete Boolean algebra with the $λ$-c.c., then no nonprincipal ultrafilter on $\mathcal{U}$ $λ^+$-saturates any unsimple theory.
We use a (countable support) creature construction to show that consistently \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{non}(\text{NULL}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \] The same method shows the consistency of \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \]
We study the values of the higher dimensional cardinal characteristics for sets of functions $f:ω^ω\to ω^ω$ introduced by the second author. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg\mathsf{CH}$ such as the Cohen, random and Sacks models and, as a byproduct show that, with possibly one exception, for the bounding numbers there are no $\mathsf{ZFC}$ relations between them beyond those in the higher dime…
There has recently been work by multiple groups in extracting the properties associated with cardinal invariants of the continuum and translating these properties into similar analogous combinatorial properties of computational oracles. Each property yields a highness notion in the Turing degrees. In this paper we study the highness notions that result from the translation of the evasion number and its dual, the prediction number, as well as two versions of the rearrangement number. When translated appropriately, these yield four new highness notions. We will define these new notions, show som…
Yorioka [J. Symbolic Logic 67(4):1373-1384, 2002] 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 ZFC. We construct a matrix iteration of ccc posets to force that, for many ideals in that class, their associated cardinal invariants (i.e. additivity, covering, uniformity and cofinality) are pairwise different. In addition, we show that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with t…
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…
We study several cardinal characteristics of closed graphs G on compact metrizable spaces. In particular, we address the question when it is consistent for the bounding number to be strictly smaller than the smallest size of a set not covered by countably many compact G-anticliques. We also provide a descriptive set theoretic characterization of the class of analytic graphs with countable coloring number.
We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak{s}_θ,\mathfrak{p}_θ,\mathfrak{g}_θ,\mathfrak{r}_θ,\mathfrak{t}_θ$ at uncountable regular cardinals $θ$. Motivated by a theorem of Raghavan-Shelah who proved that $\mathfrak{s}_θ\leq\mathfrak{b}_θ$, we explore in the first part of the paper the consistency of inequalities comparing $\mathfrak{s}_θ$ with $\mathfrak{p}_θ$ and $\mathfrak{g}_θ$. In the second part of the paper we study variations of the extender-based Radin forcing to establish several consistency results c…
A topological space $X$ is called a $Q$-space if every subset of $X$ is of type $F_σ$ in $X$. For $i\in\{1,2,3\}$ let $\mathfrak q_i$ be the smallest cardinality of a second-countable $T_i$-space which is not a $Q$-space. It is clear that $\mathfrak q_1\le\mathfrak q_2\le\mathfrak q_3$. For $i\in\{1,2\}$ we prove that $\mathfrak q_i$ is equal to the smallest cardinality of a second-countable $T_i$-space which is not perfect. Also we prove that $\mathfrak q_3$ is equal to the smallest cardinality of a submetrizable space, which is not a $Q$-space. Martin's Axiom implies that $\mathfrak q_i=\mat…
How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We define the \emph{rearrangement number}, a new cardinal characteristic of the continuum, as the answer to this question. We compare the rearrangement number with several natural variants, for example one obtained by requiring the rearranged series to still converge but to a new, finite limit. We also compare the rearrangement number with several well-studied cardinal characteristics of the continuum. We present some new…
Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong coloring principle U(...), introduced in Part I of this…
For $X$ a separable metric space define $\pp(X)$ to be the smallest cardinality of a subset $Z$ of $X$ which is not a relative $\ga$-set in $X$, i.e., there exists an $\om$-cover of $X$ with no $\ga$-subcover of $Z$. We give a characterization of $\pp(2^\om)$ and $\pp(\om^\om)$ in terms of definable free filters on $\om$ which is related to the psuedointersection number $\pp$. We show that for every uncountable standard analytic space $X$ that either $\pp(X)=\pp(2^\om)$ or $\pp(X)=\pp(\om^\om)$. We show that both of following statements are each relatively consistent with ZFC: (a) $\pp=\pp(\om…
Cichoń's diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and $σ$-compact subsets of the irrationals. It is consistent that all entries of Cichoń's diagram are pairwise different (apart from $\textrm{add}(\mathcal{M})$ and $\textrm{cof}(\mathcal{M})$, which are provably equal to other entries). However, the consistency proofs so far required large cardinal assumptions. In this work, we show the consistency without such assumptions.
A function $f:X\to Y$ between topological spaces is called $σ$-$continuous$ (resp. $\barσ$-$continuous$) if there exists a (closed) cover $\{X_n\}_{n\inω}$ of $X$ such that for every $n\inω$ the restriction $f{\restriction}X_n$ is continuous. By $\mathfrak c_σ$ (resp. $\mathfrak c_{\barσ}$) we denote the largest cardinal $κ\le\mathfrak c$ such that every function $f:X\to\mathbb R$ defined on a subset $X\subset\mathbb R$ of cardinality $|X|<κ$ is $σ$-continuous (resp. $\barσ$-continuous). It is clear that $ω_1\le\mathfrak c_{\barσ}\le\mathfrak c_σ\le\mathfrak c$. We prove that $\mathfrak p\le\m…
Technical University of Košice（科希策理工大学）；Pavol Jozef Šafárik University in Košice（科希策帕沃尔·约瑟夫·沙法里克大学）；Slovak Academy of Sciences（斯洛伐克科学院）；Technische Universität Wien（维也纳工业大学）；University of Vienna（维也纳大学）
Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $\mathfrak p(κ)=\mathfrak t(κ)$ or there is a $(\mathfrak p(κ),λ)$-gap of club-supported slaloms for some $λ< \mathfrak p(κ)$. While the existence of such gaps is unclear, this is a promising step to lift Malliaris and Shelah's proof of $\mathfrak p=\mathfrak t$ to uncountable cardinals. We do analyze gaps of slaloms and, in particular, show that $\mathfrak p(…
We introduce and analyze a new cardinal characteristic of the continuum, the \emph{splitting number of the reals}, denoted $\mathfrak{s}(\mathbb R)$. This number is connected to Efimov's problem, which asks whether every infinite compact Hausdorff space must contain either a non-trivial convergent sequence, or else a copy of $β\mathbb N$.
We try to control many cardinal characteristics by working with a notion of orthogonality between two families of forcings. We show that b^+<g is consistent
Institute of Logic, Language and Computation (ILLC), Universiteit van Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）；Amsterdam University College (AUC), Universiteit van Amsterdam（阿姆斯特丹大学学院）；Fachbereich Mathematik, Universität Hamburg（汉堡大学数学系）
In this paper we are interested in finding and evaluating cardinal characteristics of the continuum that appear in large-scale topology, usually as the smallest weights of coarse structures that belong to certain classes (indiscrete, inseparated, large) of finitary or locally finite coarse structures on $ω$. Besides well-known cardinals $\mathfrak b,\mathfrak d,\mathfrak c$ we shall encounter two new cardinals $\mathsf Δ$ and $\mathsf Σ$, defined as the smallest weight of a finitary coarse structure on $ω$ which contains no discrete subspaces and no asymptotically separated sets, respectively.…
We investigate maximal free sequences in the Boolean algebra $\mathcal{P}(ω)/\mathrm{fin}$, as defined by D. Monk. We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted $\mathfrak{f}$. Answering a question of Monk, we demonstrate the consistency of $ω_1 = \mathfrak{i} = \mathfrak{f} < \mathfrak{u} = ω_2$. In fact, this consistency is demonstrated in the model of S. Shelah for $\mathfrak{i} < \mathfrak{u}$. Our paper provides a streamlined and most…
Define the special tree number, denoted $\mathfrak{st}$, to be the least size of a tree of height $ω_1$ which is neither special nor has a cofinal branch. This cardinal had previously been studied in the context of fragments of $\mathsf{MA}$ but in this paper we look at its relation to other, more typical, cardinal characteristics. Classical facts imply that $\aleph_1 \leq \mathfrak{st} \leq 2^{\aleph_0}$, under Martin's Axiom $\mathfrak{st} = 2^{\aleph_0}$ and that $\mathfrak{st} = \aleph_1$ is consistent with $\mathsf{MA}({\rm Knaster}) + 2^{\aleph_0} = κ$ for any regular $κ$ thus the value …
We introduce a model of infinitary computation which enhances the infinite time Turing machine model slightly but in a natural way by giving the machines the capability of detecting cardinal stages of computation. The computational strength with respect to ITTMs is determined to be precisely that of the strong halting problem and the nature of the new characteristic ordinals (clockable, writable, etc.) is explored.
