Uncountable Cardinals Research Articles

On singular stationarity I (mutual stationarity and ideal-based methods)

We study several ideal-based constructions in the context of singular stationarity. By combining methods of strong ideals, supercompact embeddings, and Prikry-type posets, we obtain three consistency results concerning mutually stationary sets, and answer a well-known question of Foreman and Magidor ([7]) concerning stationary sequences on the first uncountable cardinals, ℵn, 1≤n<ω.

UNCOUNTABLE TREES AND COHEN -REALS

AbstractWe investigate some versions of amoeba for tree-forcings in the generalized Cantor and Baire spaces. This answers [10, Question 3.20] and generalizes a line of research that in the standard case has been studied in [11], [13], and [7]. Moreover, we also answer questions posed in [3] by Friedman, Khomskii, and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show ${\bf{\Sigma }}_1^1$-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between the Ramsey and the Baire properties, in slight contrast with the standard case.

Remarks on weakly linearly Lindelöf spaces

The class of weakly linearly Lindelöf spaces was introduced and studied by Juhász, Tkachuk and Wilson in [7]. Recall that a space X is weakly linearly Lindelöf if for any family U of non-empty open subsets of X of regular uncountable cardinality κ, there exists a point x∈X such that every neighborhood of x meets κ-many elements of U. In this paper, we show that:(1) If X is a weakly linearly Lindelöf space and U is an open cover of X, then for the open cover {St2(x,U):x∈X} of X, there exists a countable subset A⊂X such that St2(A,U)‾=X;(2) Every weakly linearly Lindelöf normal metaLindelöf space is weakly Lindelöf;(3) If X is a first countable regular space, then M(X) (generated by Moore Machine) is weakly linearly Lindelöf if and only if X is weakly linearly Lindelöf;(4) Every product of a weakly linearly Lindelöf space and a space of countable spread (or a separable space) is weakly linearly Lindelöf;(5) If a subspace X⊂ω1ω is weakly linearly Lindelöf, then X is second countable (and hence, metrizable);(6) If X is a weakly linearly Lindelöf Baire space with a rank 2-diagonal such that we(X)≤ω1, then |X|≤c;(7) The space X is cellular-WLL if and only if it is weakly linearly Lindelöf.

The spectrum of independence

We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote hbox {Spec}(mif). Here mif abbreviates maximal independent family. We show that:whenever kappa _1<cdots <kappa _n are finitely many regular uncountable cardinals, it is consistent that {kappa _i}_{i=1}^nsubseteq hbox {Spec}(mif);whenever kappa has uncountable cofinality, it is consistent that hbox {Spec}(mif)={aleph _1,kappa =mathfrak {c}}. Assuming large cardinals, in addition to (1) above, we can provide that (κi,κi+1)∩Spec(mif)=∅\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (\\kappa _i,\\kappa _{i+1})\\cap \\hbox {Spec}(mif)=\\emptyset \\end{aligned}$$\\end{document}for each i, 1le i<n.

UNIVERSAL CLASSES NEAR ${\aleph _1}$

AbstractShelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $-sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal:Theorem. Assume ${2^{{\aleph _0}}} &lt; {2^{{\aleph _1}}}$. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $-sentence.(1)If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left( {\psi ,\aleph _1 } \right) &lt; 2^{\aleph _1 } $, then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.(2)If ψ is categorical in ${\aleph _1}$, then ψ is categorical in all uncountable cardinals.The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $-definable tame abstract elementary classes with primes.

Measurable cardinals and good ‐wellorderings

AbstractWe study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals κ with the property that the collection of all initial segments of the wellordering is definable by a Σ1‐formula with parameter κ. A short argument shows that the existence of a measurable cardinal δ implies that such wellorderings do not exist at δ‐inaccessible cardinals of cofinality not equal to δ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that imposes restrictions on the existence of such wellorderings at uncountable cardinals. Finally, we generalise the above result to the minimal model containing two measurable cardinals.

On the existence of $$\kappa $$ κ -existentially closed groups

We prove that a $$\kappa $$ -existentially closed group of cardinality $$\lambda $$ exists whenever $$\kappa \le \lambda $$ are uncountable cardinals with $$\lambda ^{<\kappa }=\lambda $$ . In particular, we show that there exists a $$\kappa $$ -existentially closed group of cardinality $$\kappa $$ for regular $$\kappa $$ with $$2^{<\kappa }=\kappa $$ . Moreover, we prove that there exists no $$\kappa $$ -existentially closed group of cardinality $$\kappa $$ for singular $$\kappa $$ . Assuming the generalized continuum hypothesis, we completely determine the cardinals $$\kappa \le \lambda $$ for which a $$\kappa $$ -existentially closed group of cardinality $$\lambda $$ exists.

SOME APPLICATIONS OF SUPERCOMPACT EXTENDER BASED FORCINGS TO HOD

AbstractSupercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V. In particular, a model where all regular uncountable cardinals are measurable in HOD is constructed.

A Note on Hindman-Type Theorems for Uncountable Cardinals

Recent results of Hindman, Leader and Strauss and of Fernandez-Breton and Rinot showed that natural versions of Hindman’s Theorem fail for all uncontable cardinals. On the other hand, Komjath proved a result in the positive direction, showing that there are arbitrarily large abelian groups satisfying some Hindman-type property. In this note we show how a family of natural Hindman-type theorems for uncountable cardinals can be obtained by adapting some recent results of the author from their original countable setting.

Distributive proper forcing axiom and a left-right dichotomy of Cichoń’s diagram

In this paper, we study distributive proper forcing axiom (DPFA) and prove its consistency with a dichotomy of the Cichon’s diagram, relative to certain large cardinal assumption. Namely, we evaluate the cardinal invariants in Cichon’s diagram with the first two uncountable cardinals in the way that the left-hand side has the least possible cardinality while the right-hand side has the largest possible value, and preserve the evaluation along the way of forcing DPFA.

Weakly linearly lindelöf monotonically normal spaces are lindelöf

A space X is weakly linearly Lindelöf if for any family U of non-empty open subsets of X of regular uncountable cardinality κ, there exists a point x ∈ X such that every neighborhood of x meets κ-many elements of U. We also introduce the concept of almost discretely Lindelöf spaces as the ones in which every discrete subspace can be covered by a Lindelöf subspace. We prove that, in addition to linearly Lindelöf spaces, both weakly Lindelöf spaces and almost discretely Lindelöf spaces are weakly linearly Lindelöf. The main result of the paper is formulated in the title. It implies that every weakly Lindelöf monotonically normal space is Lindelöf, a result obtained earlier in [3]. We show that, under the hypothesis 2ω &lt; ωω, if the co-diagonal ΔcX = (X × X) \ΔX is discretely Lindelöf, then X is Lindelöf and has a weaker second countable topology; here ΔX = {(x, x): x ∈ X} is the diagonal of the space X. Moreover, discrete Lindelöfness of ΔcX together with the Lindelöf Σ-property of X imply that X has a countable network.

Strong failures of higher analogs of Hindman’s theorem

We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that for every $X\subseteq\mathbb R$ with $|X|=|\mathbb R|$, and every colour $\gamma\in\mathbb Q$, there are two distinct elements $x_0,x_1$ of $X$ for which $c(x_0+x_1)=\gamma$. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah. Theorem 2: For every Abelian group $G$, there exists a colouring $c:G\rightarrow\mathbb Q$ such that for every uncountable $X\subseteq G$, and every colour $\gamma$, for some large enough integer $n$, there are pairwise distinct elements $x_0,\ldots,x_n$ of $X$ such that $c(x_0+\cdots+x_n)=\gamma$. In addition, it is consistent that the preceding statement remains valid even after enlarging the set of colours from $\mathbb Q$ to $\mathbb R$. Theorem 3: Let $\circledast_\kappa$ assert that for every Abelian group $G$ of cardinality $\kappa$, there exists a colouring $c:G\rightarrow G$ such that for every positive integer $n$, every $X_0,\ldots,X_n \in[G]^\kappa$, and every $\gamma\in G$, there are $x_0\in X_0,\ldots, x_n\in X_n$ such that $c(x_0+\cdots+x_n)=\gamma$. Then $\circledast_\kappa$ holds for unboundedly many uncountable cardinals $\kappa$, and it is consistent that $\circledast_\kappa$ holds for all regular uncountable cardinals $\kappa$.

A parallel to the null ideal for inaccessible $$\lambda $$ λ : Part I

It is well known how to generalize the meagre ideal replacing $$\aleph _0$$ by a (regular) cardinal $$\lambda > \aleph _0$$ and requiring the ideal to be $$({<}\lambda )$$ -complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing $$\aleph _0$$ by $$\lambda $$ . So naturally, to call it a generalization we require it to be $$({<}\lambda )$$ -complete and $$\lambda ^+$$ -c.c. and more. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead of forcing) we may look at the Boolean Algebra of $$\lambda $$ -Borel sets modulo the ideal. Common wisdom have said that there is no such thing because we have no parallel of Lebesgue integral, but here surprisingly first we get a positive $$=$$ existence answer for a generalization of the null ideal for a “mild” large cardinal $$\lambda $$ —a weakly compact one. Second, we try to show that this together with the meagre ideal (for $$\lambda $$ ) behaves as in the countable case. In particular, we consider the classical Cichoń diagram, which compares several cardinal characterizations of those ideals. We shall deal with other cardinals, and with more properties of related forcing notions in subsequent papers (Shelah in The null ideal for uncountable cardinals; Iterations adding no $$\lambda $$ -Cohen; Random $$\lambda $$ -reals for inaccessible continued; Creature iteration for inaccesibles. Preprint; Bounding forcing with chain conditions for uncountable cardinals) and Cohen and Shelah (On a parallel of random real forcing for inaccessible cardinals. arXiv:1603.08362 [math.LO]) and a joint work with Baumhauer and Goldstern.

A generalized Borel-reducibility counterpart of Shelah’s main gap theorem

We study the $$\kappa $$ź-Borel-reducibility of isomorphism relations of complete first order theories in a countable language and show the consistency of the following: For all such theories T and $$T^{\prime }$$Tź, if T is classifiable and $$T^{\prime }$$Tź is not, then the isomorphism of models of $$T^{\prime }$$Tź is strictly above the isomorphism of models of T with respect to $$\kappa $$ź-Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations are considered on models of some fixed uncountable cardinality obeying certain restrictions.

A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals

We consider the following dichotomy for $\Sigma^0_2$ finitary relations $R$ on analytic subsets of the generalized Baire space for $\kappa$: either all $R$-independent sets are of size at most $\kappa$, or there is a $\kappa$-perfect $R$-independent set. This dichotomy is the uncountable version of a result found in (W. Kubi\'s, Proc. Amer. Math. Soc. 131 (2003), no 2.:619--623) and in (S. Shelah, Fund. Math. 159 (1999), no. 1:1--50). We prove that the above statement holds assuming $\Diamond_\kappa$ and the set theoretical hypothesis $I^-(\kappa)$, which is the modification of the hypothesis $I(\kappa)$ suitable for limit cardinals. When $\kappa$ is inaccessible, or when $R$ is a closed binary relation, the assumption $\Diamond_\kappa$ is not needed. We obtain as a corollary the uncountable version of a result by G. S\'agi and the first author (Log. J. IGPL 20 (2012), no. 6:1064--1082) about the $\kappa$-sized models of a $\Sigma^1_1(L_{\kappa^+\kappa})$-sentence when considered up to isomorphism, or elementary embeddability, by elements of a $K_\kappa$ subset of ${}^\kappa\kappa$. The role of elementary embeddings can be replaced by a more general notion that also includes embeddings, as well as the maps preserving $L_{\lambda\mu}$ for $\omega\leq\mu\leq\lambda\leq\kappa$ and the finite variable fragments of these logics.

Two inequalities between cardinal invariants

We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $\omega$ of asymptotic density $0$. We obtain an upper bound on the $\ast$-covering number, sometimes also called the weak covering number, of this ideal by proving in Section \ref{sec:covz0} that ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0}) \leq \mathfrak{d}$. In Section \ref{sec:skbk} we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We prove in sharp contrast to the case when $\kappa = \omega$, that if $\kappa$ is any regular uncountable cardinal, then ${\mathfrak{s}}_{\kappa} \leq {\mathfrak{b}}_{\kappa}$.

Hindman’s Theorem is only a Countable Phenomenon

We pursue the idea of generalizing Hindman's Theorem to uncountable cardinalities, by analogy with the way in which Ramsey's Theorem can be generalized to weakly compact cardinals. But unlike Ramsey's Theorem, the outcome of this paper is that the natural generalizations of Hindman's Theorem proposed here tend to fail at all uncountable cardinals.

A problem of Laczkovich: How dense are set systems with no large independent sets?

We investigate the function LH(n)=max⁡{|H∩P(A)|:|A|=n} where H is a set of finite subsets of λ such that every λ-sized subset of λ has arbitrarily large subsets form H. For κ=ℵ1, lim⁡LH(n)/n→∞ and LH(n)=O(n2), and in different models of set theory, either bound can be sharp. If λ>ℵ1, LH(n)>cn2 for some c>0 and n sufficiently large. If λ is strong limit singular, then LH is superpolynomial. If κ<λ are uncountable cardinals, we call a family Hκ-dense (strongly κ-dense) in λ if every κ-sized subset of λ contains a set (arbitrarily large sets) in H. We show under GCH that if H is κ+-dense in κ+r (r finite), then LH(n)/nr→∞ (κ=ω) and LH(n)>cnr+1 (κ>ω). We also give bounds for LH(n) when H has large chromatic or coloring number.

All uncountable cardinals in the Gitik model are almost Ramsey and carry Rowbottom filters

Using the analysis developed in our earlier paper , we show that every uncountable cardinal in Gitik's model of in which all uncountable cardinals are singular is almost Ramsey and is also a Rowbottom cardinal carrying a Rowbottom filter. We assume that the model of is constructed from a proper class of strongly compact cardinals, each of which is a limit of measurable cardinals. Our work consequently reduces the best previously known upper bound in consistency strength for the theory + “All uncountable cardinals are singular” + “Every uncountable cardinal is both almost Ramsey and a Rowbottom cardinal carrying a Rowbottom filter”.

Regularity properties on the generalized reals

We investigate regularity properties derived from tree-like forcing notions in the setting of “generalized descriptive set theory”, i.e., descriptive set theory on κκ and 2κ, for regular uncountable cardinals κ.