arXiv subjects
Publications and source records attributed to Saharon Shelah.
We deal with the combinatorial principle Weak Diamond, showing that we always either a local version is not saturated or we can increase the number of colours. Then we point out a model theoretic consequence of Weak Diamond.
This is a revised version (of late 2020) of [Sh:700], which is arXiv:math/0012170 . First point is noting that the proof of Theorem 4.3 in [Sh:700], which says that the proof giving the consistency $ \mathfrak{b} = \mathfrak{d} = \mathfrak{u} < \mathfrak{a} $ also gives $ \mathfrak{s} = \mathfrak{d} $. The proof uses a measurable cardinal and a c.c.c. forcing so it gives large $ \mathfrak{d} $ and assumes a large cardinal. Second point is adding to the results of \S2,\S3 which say that (in \S3 with no large cardinals) we can force $ {\aleph_1} < \mathfrak{b} = \mathfrak{d} < \mathfrak{a}$. We like to have $ {\aleph_1} < \mathfrak{s} \le \mathfrak{b} = \mathfrak{d} < \mathfrak{a} $. For this we allow in \S2,\S3 the sets $ K_t $ to be uncountable; this requires non-essential changes. In particular, we replace usually $ {\aleph_0}, {\aleph_1} $ by $ σ, \partial $. Naturally we can deal with $ \mathfrak{i} $ and similar invariants. Third we proofread the work again. To get $ \mathfrak{s} $ we could have retained the countability of the member of the $ I_t$-s but the parameters would change with $ A \in I_t$, well for a cofinal set of them; but the present seems simpler. We intend to continue in [Sh:F2009].
We expand the results of Roslanowski and Shelah arXive:1806.06283 , arXive:1909.00937 to all perfect Abelian Polish groups $(H,+)$. In particular, we show that if $α<ω_1$ and $4\leq k<ω$, then there is a ccc forcing notion adding a $Σ^0_2$ set $B\subseteq H$ which has $\aleph_α$ many pairwise $k$--overlapping translations but not a perfect set of such translations.
We find a logic really stronger than first order for the random graph with edge probability $\frac 12$ but satisfies the 0-1 law. This means that on the one hand it satisfies the 0-1 law, e.g. for the random graph ${\mathcal G}_{n,1/2}$ and on the other hand there is a formula $φ(x)$ such that for no first order $ψ(x)$ do we have: for every random enough ${\mathcal G}_{n,1/2}$ the formulas $φ(x),ψ(x)$ equivalent in it.
Fix a ring $ R $ and look at the class of left $ R $-modules and naturally, we restrict ourselves to the case of rings such that this class is not too similar to the case $ R $ is a field. We shall solve Kaplansky test problems, all three of which say that we do not have decomposition theory, e.g., if the square of one module is isomorphic to the cube of another it does not follows that they are isomorphic. Our results are in the ordinary et theory, ZFC. For this, we look at bimodules, i.e., the structures which are simultaneously left $ R $-modules and right $ S $-modules with reasonable associativity, over a commutative ring $ T $ included in the centers of $ R $ and of $ S $. Eventually, we shall choose $ S $ to help solve each of the test questions. But first, we analyze what can be the smallest endomorphism ring of an $(R, S)$-bimodule. We construct such bimodules by using a simple version of the black box.
As defined in [1], a Hausdorff space is strongly anti-Urysohn (in short: SAU) if it has at least two non-isolated points and any two infinite} closed subsets of it intersect. Our main result answers the two main questions of [1] by providing a ZFC construction of a locally countable SAU space of cardinality $2^{\mathfrak{c}}$. The construction hinges on the existence of $2^{\mathfrak{c}}$ weak P-points in $ω^*$, a very deep result of Ken Kunen. It remains open if SAU spaces of cardinality $> 2^{\mathfrak{c}}$ could exist, while it was shown in [1] that $2^{2^{\mathfrak{c}}}$ is an upper bound. Also, we do not know if crowded SAU spaces, i.e. ones without any isolated points, exist in ZFC but we obtained the following consistency results concerning such spaces. (1) It is consistent that $\mathfrak{c}$ is as large as you wish and there is a locally countable and crowded SAU space of cardinality $\mathfrak{c}^+$. (2) It is consistent that both $\mathfrak{c}$ and $2^\mathfrak{c}$ are as large as you wish and there is a crowded SAU space of cardinality $2^\mathfrak{c}$. (3) For any uncountable cardinal $κ$ the following statements are equivalent: (i) $κ=cof({[κ]}^{ω},\subseteq)$. (ii) There is a locally countable and crowded SAU space of size $κ$ in the generic extension obtained by adding $κ$ Cohen reals. (iii) There is a locally countable and countably compact $T_1$-space of size $κ$ in some CCC generic extension. [1] I. Juhasz, L. Soukup, and Z. Szentmiklossy, Anti-Urysohn spaces, Top. Appl., 213 (2016), pp. 8--23.
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 $k+1$-Knaster; and MA for the union of all $k$-Knaster forcings from MA for precaliber.
We get a quite maximal version of the colouring property $Pr_1$ by proving $Pr_1(λ,λ,λ,θ)$ when $λ= \partial^+, \partial > θ$ are regular cardinals.
We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give criteria for a theory to have singular compactness.
We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than $\beth_2(\aleph_0)$ then $G$ contains all finite subgraphs of Sh$_n(ω)$ and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability.
We prove that if $G=(V,E)$ is an $ω$-stable (respectively, superstable) graph with $χ(G)>\aleph_0$ (respectively, $2^{\aleph_0}$) then $G$ contains all the finite subgraphs of the shift graph $\text{Sh}_n(ω)$ for some $n$. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if $G$ is $ω$-stable with $\mathrm{U}(G)\leq 2$ we prove that $n\leq 2$ suffices.
We generalize the Hart-Shelah example \cite{HaSh:323} to higher infinitary logics. We build, for each natural number $k\geq 2$ and for each infinite cardinal $λ$, a sentence $ψ_k^λ$ of the logic $L_{(2^λ)^+,ω}$ that (modulo mild set theoretical hypotheses around $λ$ and assuming $2^λ< λ^{+m}$) is categorical in $λ^+,\dots,λ^{+k-1}$ but not in $\beth_{k+1}(λ)^+$ (or beyond); we study the dimensional encoding of combinatorics involved in the construction of this sentence and study various model-theoretic properties of the resulting abstract elementary class ${\mathcal K}^*(λ,k)=(Mod(ψ_k^λ),\prec_{(2^λ)^+,ω})$ in the finite interval of cardinals $λ,λ^+,\dots,λ^{+k}$.
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.
It was proved recently that Telgársky's conjecture, which concerns partial information strategies in the Banach-Mazur game, fails in models of $\mathsf{GCH}+\square$. The proof introduces a combinatorial principle that is shown to follow from $\mathsf{GCH}+\square$, namely: $\triangledown$: Every separative poset $\mathbb P$ with the $κ$-cc contains a dense sub-poset $\mathbb D$ such that $|\{ q \in \mathbb D \,:\, p \text{ extends } q \}| < κ$ for every $p \in \mathbb P$. We prove this principle is independent of $\mathsf{GCH}$ and $\mathsf{CH}$, in the sense that $\triangledown$ does not imply $\mathsf{CH}$, and $\mathsf{GCH}$ does not imply $\triangledown$ assuming the consistency of a huge cardinal. We also consider the more specific question of whether $\triangledown$ holds with $\mathbb P$ equal to the weight-$\aleph_ω$ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of $\mathsf{ZFC}+\mathsf{GCH}$.
A wide Aronszajn tree is a tree of size and height $ω_1$ with no uncountable branches. We prove that under $MA(ω_1)$ there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and Väänänen from 1994. We also prove that under the same assumption there is no universal Aronszajn tree, improving a result of Todorčevi{ć} from 2007 who proved the same under the assumption of BPFA for posets of size $\aleph_1$. Finally, we prove that under $MA(ω_1)$, every wide Aronszajn tree weakly embeds in an Aronszajn tree.
We note that some form of the condition "$p_1, p_2$ have a $\leq_{\mathbb{Q}}$-lub in $\mathbb{Q}$" is necessary in some forcing axiom for $λ$-complete $μ^+$-c.c. forcing notions. We also show some versions are really stronger than others.
This paper is a continuation of [BaSh], where we studied the behaviour of the abelianization functor under inverse limits. Our main result in [BaSh] was that if $\mathcal{T}$ is a countable directed poset and $G:\mathcal{T}\to\mathcal{G} rp$ is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$\mathrm{Ab}({\lim}_{t\in\mathcal{T}}G_t)\to {\lim}_{t\in\mathcal{T}}\mathrm{Ab}(G_t)$$ is surjective, and its kernel is a cotorsion group. The abelianization is an example of a left adjoint functor from groups to abelian groups. In this paper we study the behaviour under inverse limits of left adjoint functors from pointed sets to abelian groups. Such functors are classified by abelian groups, where to the abelian group $A$ corresponds the left adjoint functor $L_A:\mathcal{S} \text{et}_*\to\mathcal{A} \text{b}$ given by $L_A(Y)=\bigoplus_{Y\setminus\{*\}}A.$ If $\mathcal{T}$ is a directed poset and $X:\mathcal{T}\to\mathcal{S} \text{et}_*$ is a is diagram of pointed sets, we show that the natural map $$ρ:L_A({\lim}_{t\in\mathcal{T}}X_t) \to{\lim}_{t\in\mathcal{T}}L_A(X_t)$$ is injective. If, in addition, $\mathcal{T}$ is countable and $X$ satisfies the Mittag-Leffler condition, we show that the cokernel of $ρ$ is an algebraically compact group. Compared with the main result in [BaSh], algebraically compact is much stronger then cotorsion as it also requires the Ulm length to be $\leq 1$. We also show that this result, even in its weak form of cotorsion, does not extend to uncountable diagrams. Namely, if $A$ is not the product of a divisible group and a bounded group, we construct a directed poset $\mathcal{T}$ with $|\mathcal{T}|=2^{\aleph_0}$ and a diagram $X:\mathcal{T}\to\mathcal{S} \text{et}_*$, that satisfies the Mittag-Leffler condition, such that the cokernel of $ρ$ is not cotorsion.
We construct, in $\mathsf{ZFC}$, a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences, answering an old problem of van Douwen. As a consequence we also prove the existence of two countably compact groups $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ such that the product $\mathbb{G}_{0} \times \mathbb{G}_{1}$ is not countably compact, thus answering a classical problem of Comfort.