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 dimensional Cichoń diagram. In the case of the dominating numbers we show that in fact they collapse in the sense that modding out by the ideal does not change their values. Moreover, they are closely related to the dominating numbers $\mathfrak{d}^λ_κ$.
Full reflection of stationary sets at regular cardinals
一个正则不可数基数Kappa的平稳子集S在正则基数上全反射，如果对每个由正则基数组成的高阶平稳集Tsubseteq Kappa，T中存在一个α，使得S cap α是α的平稳子集。我们证明了全反射公理，即每个平稳集在正则基数上全反射，以及n-Mahlo基数的存在性与Pi^1_n-不可描述基数的存在性是等相容的.我们还对极大Mahlo基数给出了适当的推广。
We define a generic Vopěnka cardinal to be an inaccessible cardinal $κ$ such that for every first-order language $\mathcal{L}$ of cardinality less than $κ$ and every set $\mathscr{B}$ of $\mathcal{L}$-structures, if $|\mathscr{B}| = κ$ and every structure in $\mathscr{B}$ has cardinality less than $κ$, then an elementary embedding between two structures in $\mathscr{B}$ exists in some generic extension of $V$. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of $\aleph_1$-Suslin sets of reals in models of ZF. In particular, we show tha…
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.
We show that the statement "every universally Baire set of reals has the perfect set property" is equiconsistent modulo ZFC with the existence of a cardinal that we call a virtually Shelah cardinal. These cardinals resemble Shelah cardinals but are much weaker: if $0^\sharp$ exists then every Silver indiscernible is virtually Shelah in $L$. We also show that the statement $\text{uB} = {\bfΔ}^1_2$, where $\text{uB}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $Σ_2$-reflecting virtually Shelah cardinal.
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…
Assuming that $0^\dagger$ does not exist, we prove that if there is a partition of $\mathbb R$ into $\aleph_ω$ Borel sets, then there is also a partition of $\mathbb R$ into $\aleph_{ω+1}$ Borel sets.
We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.
It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact cardinal that there is a uniform ultrafilter on ${\aleph}_{ω+1}$ which is generated by fewer than ${2}^{{\aleph}_{ω+1}}$ sets.
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+…
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 $\aleph_1.$
Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses so strong as to contradict the Axiom of Choice -- we solve several variants of the generalized continuum problem and identify structural features of the levels $V_α$ of the cumulative hierarchy of sets that are eventually periodic, alternating according to the parity of the ordinal $α$. For example, if there is an elementary embedding from the universe of sets to itself, then for sufficiently large ordinals $α$, the supremum of the lengths of all wellfounded relations on $V_α$ is a strong limit cardinal if and only if…
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.
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 give a complete characterization of the sets of cardinals that in a suitable forcing extension can be the Kurepa spectrum, that is, the set of cardinalities of branches of Kurepa trees. This answers a question of the first named author.
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 establish some similarities/analogies between uncountable cardinals or powersets and the class $V$ of all sets. They concern mainly the Boolean algebras ${\cal P}(κ)$, for a regular cardinal $κ$, and ${\cal C}(V)$ (the class of subclasses of the universe $V$), endowed with some ideals, especially the ideal $[κ]^{<κ}$ for ${\cal P}(κ)$, and the ideal of sets $V$ for ${\cal C}(V)$.
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…
