A function f from reals to reals (f:R->R) is almost continuous (in the sense of Stallings) iff every open set in the plane which contains the graph of f contains the graph of a continuous function.
Natkaniec showed that for any family F of continuum many real functions there exists g:R->R such that f+g is almost continuous for every f in F. Let AA be the smallest cardinality of a family F of real functions for which there is no g:R->R with the property that f+g is almost continuous for every f in F. Thus Natkaniec showed that AA is strictly greater than the continuum. He asked if anything more could be said.
We show that the cofinality of AA is greater than the continuum, c. Moreover, we show that it is pretty much all that can be said about AA in ZFC, by showing that AA can be equal to any regular cardinal between c^+ and 2^c (with 2^c arbitrarily large). We also show that AA = AD where AD is defined similarly to AA but for the class of Darboux functions. This solves another problem of Maliszewski and Natkaniec.
The present status of the problems in my book "Cardinal Invariants on Boolean algebras" (Birkhauser 1996) is described, with a description of solutions or partial solutions, and references.
I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length $\sup\set{j(f)(κ) \st f: κ\to κ}$. Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, \Vopenka\ cardinals, hypercompact cardinals, a…
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…
We prove several consistency results concerning the notion of $ω$-strongly measurable cardinal in HOD. In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(κ) = κ$, that every successor of a regular cardinal is $ω$-strongly measurable in HOD.
We formulate and prove (in {\sf ZFC}) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle $Pr_1(μ^+,μ^+,μ^+,\cf(μ))$ for singular $μ$.
Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal $\mathfrak r$. This is achieved through analyzing nowhere reaping families of subsets of $\mathbb N$ and through applying a paving lemma which is a consequence of a p…
Assume G.C.H. and kappa is the first uncountable cardinal such that there is a kappa-free abelian group which is not a Whitehead (abelian) group. We prove that kappa is necessarily an inaccessible cardinal
Through careful analysis of an argument of Brooke-Taylor and Rosicky, we show that the powerful image of any accessible functor is closed under colimits of $κ$-chains, $κ$ a sufficiently large almost measurable cardinal. This condition on powerful images, by methods resembling those of Lieberman and Rosicky, implies $κ$-locality of Galois types. As this, in turn, implies sufficient measurability of $κ$, via a paper of Boney and Unger, we obtain an equivalence: a purely category-theoretic characterization of almost measurable cardinals.
We strengthen non-structure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fraisse games between a fixed group of cardinality lambda and a free Abelian group. A group is called epsilon-game-free if the isomorphism player has a winning strategy in the game (of the described form) of length epsilon in lambda. We prove for a large set of successor cardinals lambda=mu^+ existence of nonfree (mu*omega_1)-game-free groups of cardinality lambda. We concentrate on successors of singular cardinals.
We solve four out of the six open problems concerning critical cardinalities of topological diagonalization properties involving tau-covers, show that the remaining two cardinals are equal, and give a consistency result concerning this remaining cardinal. Consequently, 21 open problems concerning potential implications between these properties are settled. We also give structural results based on the combinatorial techniques.
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.
After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals.
We lay the combinatorial foundations for [ShSt:340] by setting up and proving the essential properties of the coding apparatus for singular cardinals. We also prove another result concerning the coding apparatus for inaccessible cardinals.
We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality $\mathfrak p\leq\mathfrak b$ does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting numbers. Lastly, we show that the inclusion ordering on $\mathcal P(ω)…
The almost disjointness numbers associated to the quotients determined by the transfinite products of the ideal of finite sets are investigated. A $\mathrm{ZFC}$ lower bound involving the minimum of the classical almost disjointness and splitting numbers is proved for these characteristics. En route, it is shown that the splitting numbers associated to these quotients are all equal to the classical splitting number. Finally, it is proved to be consistent that the almost disjointness numbers associated to these quotients are all equal to the second uncountable cardinal while the bounding number…
We prove some consistency results about b(lambda) and d(lambda), which are natural generalisations of the cardinal invariants of the continuum b and d. We also define invariants b_cl(lambda) and d_cl(lambda), and prove that almost always b(lambda) = b_cl(lambda) and d(lambda)=d_cl(lambda)
The notions of almost everywhere (a.e.) domination and its uniform version were introduced and studied in reverse mathematics. This paper studies these notions from a recursion-theoretic point of view and explore their connections to notions such as randomness and genericity. It is shown that if $Z$ is a.e. dominating then each $1$-$Z$-random is $2$-random. In other words, $0'\leq_{\rm LR} Z$ for every a.e. dominating $Z$, where ${\rm LR}$ denotes low-for-random reducibility. Other results and corollaries are also given.
We begin with the existence of groups with trivial duals for cardinals aleph_n (n in omega). Then we derive results about strongly aleph_n-free abelian groups of cardinality aleph_n (n in omega) with prescribed free, countable endomorphism ring. Finally we use combinatorial results of [Sh:108], [Sh:141] to give similar answers for cardinals >aleph_omega. As in Magidor and Shelah [MgSh:204], a paper concerned with the existence of kappa-free, non-free abelian groups of cardinality kappa, the induction argument breaks down at aleph_omega. Recall that aleph_omega is the first singular cardinal an…
We introduce 0^h (0^handgrenade) as a sharp for an inner model with a proper class of strong cardinals. If 0^h does not exist then any normal iteration tree is "almost linear." We exploit this fact to prove the existence of the core model K in the theory "ZFC + 0^h does not exist." (As of today, non-0^h is thereby the weakest anti large cardinal assumption under which K can be shown to exist in ZFC. In this sense we improve earlier work of Dodd, Jensen, Mitchell, and - partially - Steel.) We indicate that our paper provides the last step for determining the exact consistency strength of a stat…
The relations M(kappa,lambda,mu)->B [resp. B(sigma)] meaning that if A subset [kappa]^lambda with |A|=kappa is mu-almost disjoint then A has property B [resp. has a sigma-transversal] had been introduced and studied under GCH by Erdos and Hajnal in 1961. Our two main results here say the following: Assume GCH and rho be any regular cardinal with a supercompact [resp. 2-huge] cardinal above rho. Then there is a rho-closed forcing P such that, in V^P, we have both GCH and M(rho^{(+rho+1)},rho^+,rho) not-> B [resp. M(rho^{(+rho+1)},lambda,rho) not-> B(rho^+) for all lambda =< rho^{(+rho+1)}].
J. Zapletal asked if all the forcing notions considered in his monograph are homogeneous. Specifically, he asked if the forcing consisting of Borel sets of $σ$-finite 2-dimensional Hausdorff measure in $\mathbb{R}^3$ (ordered under inclusion) is homogeneous. We give a partial negative answer to both questions by showing that this $σ$-ideal is not homogeneous. Let $\mathcal{N}^1_2$ be the $σ$-ideal of sets in the plane of 1-dimensional Hausdorff measure zero. D. H. Fremlin determined the position of the cardinal invariants of this $σ$-ideal in the Cichoń Diagram. This required proving numerous …
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 continuum. Even more, Sacks forcing can be used to force that $\mathrm{non}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$, which is the first consistency result where more than two cardinal invariants associated with $\mathcal…
We show that from a supercompact cardinal κ, there is a forcing extension V[G] that has a symmetric inner model N in which ZF + not AC holds, κ and κ^+ are both singular, and the continuum function at κ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of κ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of ZF + not AC_ω in which either (1) aleph_1 and aleph_2 are both singular and the continuum function at aleph_1 can be precisely controlled, or (2) aleph_ω and aleph_{ω+…
Given a family $F$ of pairwise almost disjoint sets on a countable set $S$, we study maximal almost disjoint (mad) families $F^+$ extending $F$. We define $a^+(F)$ to be the minimal possible cardinality of $F^+\setminus F$ for such $F^+$, and $a^+(κ)=\sup\{a^+(F): |F| \leq κ\}$. We show that all infinite cardinal less than or equal to the continuum continuum can be represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities $\aleph_1=a<a^+(\aleph_1)=c$ and $a=a^+(\aleph_1)<c$ are both consistent. We also give a several constructions of mad families with some additional prope…
The essentially non-free spectrum is the class of uncountable cardinals kappa in which there is an essentially non-free algebra of cardinality kappa which is almost free. In L, the essentially non-free spectrum of a variety is entirely determined by whether or not the construction principle holds. In ZFC may be more complicated. For some varieties, such as groups, abelian groups or any variety of modules over a non-left perfect ring, the essentially non-free spectrum contains not only aleph_1 but aleph_n for all n>0. The reason for this being true in ZFC (rather than under some special set the…
Many of the most common reverse Easton iterations found in the large cardinal context, such as the Laver preparation, admit a gap at some small delta in the sense that they factor as P*Q, where P has size less than delta and Q is forced to be delta-strategically closed. In this paper, generalizing the Levy-Solovay theorem, I show that after such forcing, every embedding j:V[G]-->M[j(G)] in the extension which satisfies a mild closure condition is the lift of an embedding j:V-->M in the ground model. In particular, every ultrapower embedding in the extension lifts an embedding from the ground m…
We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum separable metric spaces on aleph_1 so that every separable metric space is almost isometrically embedded into one of them when the continuum hypothesis fails. (3) There is no collection of fewer than continuum metric spaces of cardinality aleph_2 so that every ultra-metric space of cardinality aleph_2 is almost isometrically embedded into one of them if aleph_2…
Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $κ=λ^+$ for some $λ\geq ω$ and $\mathfrak b(κ)=κ^+$ then $\mathfrak a_e(κ)=\mathfrak a_p(κ)=κ^+$. If, additionally, $2^{<λ}=λ$ then $\mathfrak a_g(κ)=κ^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak d(κ)$ in terms of $\mathfrak r(κ)$, including $\mathfrak d(κ)\leq \mathfrak r_σ(κ)\leq \mathrm{cof}([\mathfrak r(κ)]^ω)$, and…
We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every inaccessible cardinal $κ$, if $κ$ admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation $κ\nrightarrow[κ]^2_κ$ implies that for every Abelian group $(G,+)$ of size $κ$, there exists a map $f:G\rightarrow G$ such that, for every $X\subseteq G$ of size $κ$ and every $g\in G$, there exis…
We show that in Zermelo-Fraenkel Set Theory without the Axiom of Choice a surjectively modified continuum function $θ(κ)$ can take almost arbitrary values for all infinite cardinals. This choiceless version of Easton's Theorem is in sharp contrast to the situation in ZFC, where for singular cardinals $κ$, the value of $2^κ$ is strongly influenced by the behaviour of the continuum function below. Our construction can roughly be described as follows: In a ground model $V \models ZFC + GCH$ with a "reasonable" function $F: Card \rightarrow Card$ on the infinite cardinals, a class forcing ${\mathb…
Given an ideal $I$ on $ω$ let $a(I) $ ($\bar{a}(I)$) be minimum of the cardinalities of infinite (uncountable) maximal $I$-almost disjoint subsets of $[ω]^ω$, and denote $b_I$ and$d_I$ the unbounding and dominating numbers of $(ω^ω,\le_I)$. We show that (1) $a(I)>omega$ if $I$ is a summable ideal; (2) $a(Z)=ω$ and $\bar{a}(Z)\le a$ if $Z$ is a tall density ideal, (3) $b\le \bar{a}(I)$, and $b_I=b$ and $d_I=d$, for any analytic P-ideal $I$ on $ω$. Given an analytic $P$-ideal $I$ we investigate the relationship between the Sack, the $I$-bounding, $I$-dominating and $ω^ω$-bounding properties of a…
Two sets are said to be almost disjoint if their intersection is finite. Almost disjoint subsets of [omega]^omega and omega^omega have been studied for quite some time. In particular, the cardinal invariants a and a_e, defined to be the minimum cardinality of a maximal infinite almost disjoint family of [omega]^omega and omega^omega respectively, are known to be consistently less than continuum. Here we examine analogs for functions in R^omega and projections on l^2, showing that they too can be consistently less than continuum.
The main goal of this paper is to generalize several results concerning cardinal invariants to the statements about the associated families of sets. We also discuss the relationship between the additive properties of sets and their Borel images. Finally, we present estimates for the size of the smallest set which is not strongly meager.
We construct, under the assumption that union of less than continuum many meager subsets of R is meager in R, an additive connectivity function f:R-->R with Cantor intermediate value property which is not almost continuous. This gives a partial answer to a question of D. Banaszewski. We also show that every extendable function g:R-->R with a dense graph satisfies the following stronger version of the SCIVP property: for every a<b and every perfect set K between g(a) and g(b) there is a perfect subset C of (a,b) such that g[C] subset K and g|C is continuous strictly increasing. This property is…
We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $κ$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both $T$ and $A$ have cardinality $κ$, (2) $|\bar{a}\cap T|=κ$ for each $a\in A$, (3) for each uncountable subset of $T$ contains some elements of $A$, and so (i) there is an almost disjoint family of subsets of the reals with size and chromatic number $κ$, (ii) there is a locally compact, locally countable $T_2$ space with cardinality spectrum $\{ω,κ\}$.
Let X be an infinite set of regular cardinality. We determine all clones on X which contain all almost unary functions. It turns out that independently of the size of X, these clones form a countably infinite descending chain. Moreover, all such clones are finitely generated over the unary functions. In particular, we obtain an explicit description of the only maximal clone in this part of the clone lattice. This is especially interesting if X is countably infinite, in which case it is known that such a description cannot be obtained for the second maximal clone over the unary functions.
Using Shelah's revised GCH theorem we prove that if mu<beth_omega <= lambda are cardinals, then every mu-almost disjoint subfamily B of [lambda]^{beth_omega} is essentially disjoint, i.e. for each b from B there is a subset f(b) of b of size < |b| such that the family {b-f(b) b in B} is disjoint. We also show that if mu<=kappa<=lambda, and kappa is infinite, and (x) every mu-almost disjoint subfamily of [lambda]^kappa is essentially disjoint, then (xx) every mu-almost disjoint family B of subsets of lambda with |b|>=kappa for all b from B has a conflict-free colorings with kappa colors. Puttin…
The second author has recently shown ([20]) that any selectively (a) almost disjoint family must have cardinality strictly less than $2^{\alpeh_0}$, so under the Continuum Hypothesis such a family is necessarily countable. However, it is also shown in the same paper that $2^{\alpeh_0} < 2^{\alpeh_1}$ alone does not avoid the existence of uncountable selectively (a) almost disjoint families. We show in this paper that a certain effective parametrized weak diamond principle is enough to ensure countability of the almost disjoint family in this context. We also discuss the deductive strength of t…
