Search

An uncountable $\aleph_1$-free group can not admit a Polish group topology but an uncountable $\aleph_1$-free abelian group can, as witnessed e.g. by the Baer-Specker group $\mathbb{Z}^\omega$. In this paper we study $\aleph_1$-free abelian non-Archimedean Polish groups. We prove two results. The first is that being $\aleph_1$-free abelian is a complete co-analytic problem in the space of closed subgroups of $S_\infty$. The second is that there are continuum many $\aleph_1$-free abelian non-Archimedean Polish groups which are not topologically isomorphic to product groups and are pairwise not continuous homomorphic images of each other.

Suppose that $G$ is a graph of cardinality $\mu^+$ with chromatic number $\chi(G)\geq \mu^+$. One possible reason that this could happen is if $G$ contains a clique of size $\mu^+$. We prove that this is indeed the case when the edge relation is stable. When $G$ is a random graph (which is simple but not stable), this is not true. But still if in general the complete theory of $G$ is simple, $G$ must contain finite cliques of unbounded sizes.

A group is Artinian if there is no infinite strictly descending chain of subgroups. Ol'shanskii has asked whether there are Artinian groups of arbitrarily large cardinality. We reduce this problem to an analogous question, regarding universal algebras, asked by J\'onsson in the 1960s. We provide Artinian groups of cardinality $\aleph_n$ for each natural number $n$. We also give a consistent strong negative answer to the question of Ol'shanskii (from a large cardinal assumption) as well as a consistent positive answer. Thus, answers to the questions of Ol'shanskii and J\'onsson are independent of set theory.

Assuming G\"{o}del's axiom of constructibility $\bold V=\bold L,$ we present a characterization of those groups $L$ for which there exist arbitrarily large groups $H$ such that $aut(H) \cong L$. In particular, we show that it suffices to have one such group $H$ such that the size of its center is bigger than $ 2^{|L |+\aleph_0}$.

We introduce a new weak variation of diamond that is meant to only guess the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by proving it is compatible with Martin's axiom. We then prove that this principle is nontrivial by showing it may consistently fail.

We build models using an indiscernible model sub-structures of ${\kappa} \ge {\lambda}$ and related more complicated structures. We use this to build various Boolean algebras.

In [9] we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper we show that our construction from [9] satisfies several additional properties of interest. We deduce from this that countable torsion-free abelian groups are faithfully Borel complete, in fact, more strongly, we can $\mathfrak{L}_{\omega_1, \omega}$-interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (equiv., elementary embeddability) among countable models of $\mathrm{Th}(\mathbb{Z}^{(\omega)})$ is a complete analytic quasi-order.

We sow that there exists a generic extension of the G\"{o}del's constructible universe in which diamond holds and there exists a subset $Y \subseteq \omega_1$ such that for stationary many $\delta < \omega_1,$ the set $Y \cap \delta$ is not definable in the structure $(L_{F(\delta)}, \in)$, where $F(\delta) > \delta$ is the least ordinal such that $L_{F(\delta)}\models$``$\delta$ is countable''., Comment: This is publication number 1227 in Shelah's list

Paolini and Shelah have constructed absolutely Hopfian torsion-free abelian groups of any given size. In contrast, we show that this is not necessarily the case for absolutely co-Hopfian groups. We apply the infinitary logic to show a dichotomy behavior of co-Hopfian groups, by showing that the first beautiful cardinal is the turning point for this property. Namely, we prove that there are no absolute co-Hopfian abelian groups above the first beautiful cardinal. An extension of this result to the category of modules over a commutative ring is given.

In this paper, we deal with the notions of naturality from category theory and definablity from model theory and their interactions. In this regard, we present three results. First, we show, under some mild conditions, that naturality implies definablity. Second, by using the reverse Easton iteration of Cohen forcing notions, we construct a transitive model of ZFC in which every uniformisable construction is weakly natural. Finally, we show that if F is a natural construction on a class K of structures which is represented by some formula, then it is uniformly definable without any extra parameters. Our results answer some questions by Hodges and Shelah.

For an $\aleph_1$-categorical atomic class, we clarify the space of types over the unique model of size $\aleph_1$. Using these results, we prove that if such a class has a model of size $\beth_1^+$ then it is $\omega$-stable., Comment: 22 pages

The Ramsey Choice principle for families of $n$-element sets, denoted $\mathrm{RC}_n$, states that every infinite set $X$ has an infinite subset $Y\subseteq X$ with a choice function on $[Y]^n := \{z\subseteq Y : |z| = n\}$. We investigate for which positive integers $m$ and $n$ the implication $\mathrm{RC}_m \Rightarrow \mathrm{RC}_n$ is provable in ZF. It will turn out that beside the trivial implications $\mathrm{RC}_m \Rightarrow \mathrm{RC}_m$, under the assumption that every odd integer $n>5$ is the sum of three primes (known as ternary Goldbach conjecture), the only non-trivial implication which is provable in ZF is $\mathrm{RC}_2 \Rightarrow \mathrm{RC}_4$., Comment: 9 pages. arXiv admin note: text overlap with arXiv:2101.07840

Good frames were suggested in [Sh:h] as the (bare-bones) parallel, in the context of AECs, to superstable (among elementary classes). Here we consider $(\mu,\lambda,\kappa)$-frames as candidates for being (in the context of AECs) the correct parallel to the class of $|T|^+$-saturated models of a strictly stable theory (among elementary classes). One thing we lose compared to the superstable case is that going up by induction on cardinals is problematic (for stages of small cofinality). But this arises only when we try to lift such classes to higher cardinals. Also, we may use, as a replacement, the existence of prime models over unions of increasing chains. For this context we investigate the dimension., Comment: arXiv admin note: substantial text overlap with arXiv:2304.04467

The theme of the first two sections, is to prepare the framework of how from a ``complicated'' family of so called index models $I \in K_1$ we build many and/or complicated structures in a class $K_2$. The index models are characteristically linear orders, trees with $\kappa+1$ levels (possibly with linear order on the set of successors of a member) and linearly ordered graphs; for this we formulate relevant complicatedness properties (called bigness). In the third section we show stronger results concerning linear orders. If for each linear order $I$ of cardinality $\lambda > \aleph_0$ we can attach a model $M_I \in K_\lambda$ in which the linear order can be embedded such that for enough cuts of $I$, their being omitted is reflected in $M_I$, then there are $2^\lambda$ non-isomorphic cases. We also do the work for some applications., Comment: This was mistakenly submitted as new rather than an update of arXiv:1011.3576

Part I: We would like to generalize imaginary elements, weight of ${\rm ortp}(a,M,N),{\mathbf P}$-weight, ${\mathbf P}$-simple types, etc. from [Sh:c, Ch.III,V,\S4] to the context of good frames. This requires allowing the vocabulary to have predicates and function symbols of infinite arity, but it seemed that we do not suffer any real loss. Part II: become [1238] Good frames were suggested in [Sh:h] as the (bare bones) right parallel among a.e.c. to superstable (among elementary classes). Here we consider $(\mu,\lambda,\kappa)$-frames as candidates for being the right parallel to the class of $|T|^+$-saturated models of a stable theory (among elementary classes). A loss as compared to the superstable case is that going up by induction on cardinals is problematic (for cardinals of small cofinality). But this arises only when we try to lift. But this context we investigate the dimension. Part III: become [1239] In the context of Part II, we consider the main gap problem for the parallel of somewhat saturated model; showing we are not worse than in the first order case., Comment: arXiv admin note: substantial text overlap with arXiv:2304.04467

We deal with the monadic (second-order) theory of order. We prove all known results in a unified way, show a general way of reduction, prove more results and show the limitation on extending them. We prove (CH) that the monadic theory of the real order is undecidable. Our methods are model-theoretic, and we do not use automaton theory. This is a slightly corrected version of a very old work.

An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey Theorem on uncountable cardinals asserting that if we color edges of the complete graph we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hru\v{s}\'ak and Shelah are answered by showing that assuming the consistency of suitable large cardinals the following are relatively consistent with $\mathsf{ZFC}$: $\kappa\to_{hc} (\kappa)^2_\omega$ for every regular cardinal $\kappa\geq \aleph_2$ and $\neg\mathsf{CH}+ \aleph_2 \to_{hc} (\aleph_1)^2_\omega$. Building on a work of Lambie-Hanson, we also show that $\aleph_2 \to_{hc} [\aleph_2]^2_{\omega,2}$ is consistent with $\neg\mathsf{CH}$. To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals., Comment: Number 1242 on Shelah's publication list. 18 pages

Was paper 839 in the author's list until winter 2023 when it was divided into three. Part I: We would like to generalize imaginary elements, weight of ortp$(a,M,N), {\mathbf P}$-weight, ${\mathbf P}$-simple types, etc. from [She90, Ch. III,V,\S4] to the context of good frames. This requires allowing the vocabulary to have predicates and function symbols of infinite arity, but it seemed that we do not suffer any real loss. Part II: Good frames were suggested in [She09d] as the (bare bones) right parallel among a.e.c. to superstable (among elementary classes). Here we consider $(\mu, \lambda, \kappa)$-frames as candidates for being the right parallel to the class of $|T|^+$-saturated models of a stable theory (among elementary classes). A loss as compared to the superstable case is that going up by induction on cardinals is problematic (for cardinals of small cofinality). But this arises only when we try to lift. For this context we investigate the dimension. Part III: In the context of Part II, we consider the main gap problem for the parallel of somewhat saturated model; showing we are not worse than in the first order case.

We prove that for lambda = beta_omega or just lambda strong limit singular of cofinality aleph_0, if there is a universal member in the class K^lf_lambda of locally finite groups of cardinality lambda, then there is a canonical one (parallel to special models for elementary classes, which is the replacement of universal homogeneous ones and saturated ones in cardinals lambda = lambda^ < lambda). For this, we rely on the existence of enough indecomposable such groups, as proved in "Density of indecomposable locally finite groups". We also more generally deal with the existence of universal members in general classes for such cardinals., Comment: arXiv admin note: text overlap with arXiv:1901.09747

Relying on the techniques and ideas from our recent paper [13], we prove several anti-classification results for various rigidity conditions in countable abelian and nilpotent groups. We prove three main theorems: (1) the rigid abelian groups are complete co-analytic in the space of countable torsion-free abelian groups ($\mathrm{TFAB}_\omega$); (2) the Hopfian groups are complete co-analytic in $\mathrm{TFAB}_\omega$; (3) the co-Hopfian groups are complete co-analytic in the space of countable $2$-nilpotent groups. In combination with our result from [13, S5], which shows that the endo-rigid abelian groups are complete co-analytic in $\mathrm{TFAB}_\omega$, this shows that four major notions of rigidity from (abelian) group theory are as complex as possible as co-analytic problems. Further, the second and third theorem above solve two open questions of Thomas from [18], who asked this for the space of all countable groups. We leave open the question of whether the co-Hopfian mixed abelian groups are complete co-analytic in the space of countable abelian groups, but we reduce the problem to a concrete question on profinite groups, showing that if $G$ is a countable co-Hopfian abelian reduced group, then, for every prime number $p$, the torsion subgroup $\mathrm{Tor}_p(G)$ of $G$ is finite and $G$ embeds in the profinite group $ \prod_{p \in \mathbb{P}} \mathrm{Tor}_p(G)$.

We show that, consistently, there exists a Borel set B subset Cantor admitting a sequence (eta_alpha:alpha5 for any distinct eta,nu from P. This answers two questions from our previous works.

We prove that consistently, cov($\mathcal{M})< \lambda_\mathbf{0} < \lambda_\mathbf{1} < \lambda_\mathbf{\infty} < 2^{\aleph_0}$, where $\lambda_\mathbf{0}$ denotes the weak Borel chromatic number of the Kechris-Solecki-Todor\v{c}evi\'c graph $\mathbb{G}_0$, that is, the minimal cardinality of a $\mathbb{G}_0$-independent Borel covering of $2^\omega$, while $\lambda_\mathbf{1}$ and $\lambda_\infty$ are the corresponding invariants of the graph $\mathbb{G}_1$ and the simple graph associated with the equivalence relation $E_0$.

We consider $(<\lambda)$-support iterations of $(<\lambda)$-strategically complete $\lambda^+$-c.c. definable forcing notions along partial orders. We show that such iterations can be corrected to yield an analog of a result by Judah and Shelah for finite support iterations of Suslin ccc forcing, namely that if $(\mathbb{P}_{\alpha}, \underset{\sim}{\mathbb Q_{\beta}} : \alpha \leq \delta, \beta <\delta)$ is a FS iteration of Suslin ccc forcing and $U\subseteq \delta$ is sufficiently closed, then letting $\mathbb{P}_U$ be the iteration along $U$, we have $\mathbb{P}_U \lessdot \mathbb{P}_{\delta}$.

We prove that superclub implies $\mathfrak{s}=\aleph_1$. More generally, superclub at a successor of a weakly compact cardinal implies $\mathfrak{s}_\kappa=\kappa^+$. Based on this statement, we separate tiltan from superclub at a successor of a supercompact cardinal. We use Galvin's property in order to separate tiltan from superclub at successors of both regular and singular cardinals., Comment: Extended introduction, references

Let $\kappa$ be an inaccessible cardinal, $\mathfrak{U}$ be a universal algebra, and $\sim$ be the equivalence relation on $\mathfrak{U}^{\kappa}$ of eventual equality. From mild assumptions on $\kappa$ we give general constructions of $\mathcal{E} \in End(\mathfrak{U}^{\kappa}/\sim)$ satisfying $\mathcal{E} \circ \mathcal{E} = \mathcal{E}$ which do not descend from $\Delta \in End(\mathfrak{U}^{\kappa})$ having small strong supports. As an application there exists an $\mathcal{E} \in End(\mathbb{Z}^{\kappa}/\sim)$ which does not come from a $\Delta \in End(\mathbb{Z}^{\kappa})$.

For a given cardinal $\lambda$ and a torsion abelian group $K$ of cardinality less than $\lambda$, we present, under some mild conditions (for example $\lambda=\lambda^{\aleph_0}$), boundedly endo-rigid abelian group $G$ of cardinality $\lambda$ with $Tor(G)=K$. Essentially, we give a complete characterization of such pairs $(K, \lambda)$. Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existing problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals $\lambda> 2^{\aleph_{0}}$ for which there is a co-Hopfian abelian group of size $\lambda$.

We investigate the existence of metric spaces which, for any coloring with a fixed number of colors, contain monochromatic isomorphic copies of a fixed starting space K. In the main theorem we construct such a space of size \(2^{\aleph_0}\) for colorings with \(\aleph_0\) colors and any metric space \(K\) of size \(\aleph_0\). We also give a slightly weaker theorem for countable ultrametric \(K\) where, however, the resulting space has size~\(\aleph_1\).

This paper continues a line of investigation of the Halpern--L\"{a}uchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--L\"{a}uchli Theorem for one tree of height $\kappa$ holds whenever $\kappa$ is strongly inaccessible and the coloring takes less than $\kappa$ colors. We prove consistency of the Halpern--L\"{a}uchli Theorem for finitely many trees of height $\kappa$, where $\kappa$ is a strong limit cardinal of countable cofinality. On the other hand, we prove failure of weak forms of Halpern--\Lauchli\ for trees of height $\kappa$, whenever $\kappa$ is a strongly inaccessible, non-Mahlo cardinal or a singular strong limit cardinal with cofinality the successor of a regular cardinal. We also prove failure in $L$ of a weak version for all strongly inaccessible, non-weakly compact cardinals., Comment: 15 pages. This is paper 1230 on Shelah's list. Theorem 1.8 is improved to a stronger version

These are abstracts of most of the papers up to publication no. [Sh:143] (and [Sh:217]), mostly compiled in 1980/81 with R. Grossberg. Also more details than in the originals were added in [Sh:5], [Sh:8] [Sh:217], and [C2], [C3] were added (the Cxx are representations of the author's works).

We show that one can force the Measuring principle without adding any new reals. We also show that it is consistent with the large continuum. These results answer two famous questions of Justin Moore., Comment: There are some gaps in the proofs. We will soon replace a new version fixing them

We prove that it is consistent with large values of the continuum that there are no S-spaces. We also show that we can also have that compact separable spaces of countable tightness have cardinality at most the continuum.

We investigate the statement ``all automorphisms of $\mathcal P(\lambda)/[\lambda]^{<\lambda}$ are trivial''. We show that MA implies the statement for regular uncountable $\lambda<2^{\aleph_0}$; that the statement is false for measurable $\lambda$ if $2^\lambda=\lambda^+$; and that for ``densely trivial'' it can be forced (together with $2^\lambda=\lambda^{++}$) for inaccessible $\lambda$.

For a given stationary set $S$ of countable ordinals we prove (in $\mathbf{ZFC}$) that the assertion "every $S$-ladder system has $\aleph_0$-uniformization" is equivalent to "every strongly $\aleph_1$-free abelian group of cardinality $\aleph_1$ with non-freeness invariant $\subseteq S$ is $\aleph_1$-coseparable, i.e. Ext$(G, \oplus_{i=0}^{\infty} \mathbb Z)=0$ (in particular Whitehead, i.e.\ Ext$(G, \mathbb Z)=0$)". This solves problems B3 and B4 from Eklof and Mekler's monograph.

The cardinal invariants $ \mathfrak h, \mathfrak b, \mathfrak s$ of $\mathcal P (\omega)$ are known to satisfy that $\omega_1 \leq \mathfrak h \leq\min\{\mathfrak b, \mathfrak s\}$. We prove that all inequalities can be strict. We also introduce a new upper bound for $\mathfrak h$ and show that it can be less than $\mathfrak s$. The key method is to utilize finite support matrix iterations of ccc posets following \cite{BlassShelah}.

The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of $\mathbf{GCH}$ $\kappa$ is supercompact and the cardinals $\theta < \kappa$, $\lambda > \kappa$ are regular, as an application of a more general method we obtain a forcing extension in which $\textrm{cf}(\kappa) = \theta$, the Singular Cardinal Hypothesis fails at $\kappa$ and there exists a universal graph in cardinality $\lambda \in (\kappa,2^\kappa)$.

We continue the investigation started in [Sh:1215] about the relation between the Keilser-Shelah isomorphism theorem and the continuum hypothesis. In particular, we show it is consistent that the continuum hypothesis fails and for any given sequence $\mathbf m=\langle (\mathbb{M}^{1}_n, \mathbb{M}^{2}_n: n < \omega \rangle$ of models of size at most $\aleph_1$ in a countable language, if the sequence satisfies a mild extra property, then for every non-principal ultrafilter $\mathcal D$ on $\omega$, if the ultraproducts $\prod\limits_{\mathcal D} \mathbb{M}^{1}_n$ and $\prod\limits_{\mathcal D} \mathbb{M}^{2}_n$ are elementarily equivalent, then they are isomorphic., Comment: This is publication 1223 of the second author

We force the existence of a non-trivial $\kappa$-complete ultrafilter over $\kappa$ which fails to satisfy the Galvin property. This answers a question asked by the first author and Moti Gitik.

Theorem: There is a {\em complete sentence} $\phi$ of $L_{\omega_1,\omega}$ such that $\phi$ has maximal models in a set of cardinals $\lambda$ that is cofinal in the first measurable $\mu$ while $\phi$ has no maximal models in any $\chi \geq \mu$., Comment: 27 pages

This paper contains portions of Baldwin's talk at the Set Theory and Model Theory Conference (Institute for Research in Fundamental Sciences, Tehran, October 2015) and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of $L_{\omega_1,\omega}$ that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again., Comment: 26 pages

This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph $(\mu,R)$ we can find a family $\{\mathbb G_\alpha: \alpha < \mu\}$ of abelian groups such that for each $\alpha,\beta\in\mu$: $$Ext(\mathbb G_\alpha, \mathbb G_\beta) = 0 \Longleftrightarrow(\alpha,\beta) \in R.$$ In this regard, we present four results. First, we give a connection to Quillen's small object argument which helps $Ext$ vanishes and uses to present useful criteria to the question. Suppose $\lambda = \lambda^{\aleph_0}$ and $\mu = 2^\lambda$. We apply Jensen's diamond principle along with the criteria to present $\lambda$-free abelian groups representing bipartite graphs. Third, we use a version of the black box to construct in ZFC, a family of $\aleph_1$-free abelian groups representing bipartite graphs. Finally, applying forcing techniques, we present a consistent positive answer for general graphs.

For a set $M$, $\operatorname{fin}(M)$ denotes the set of all finite subsets of $M$, $M^2$ denotes the Cartesian product $M\times M$, $[M]^2$ denotes the set of all $2$-element subsets of $M$, and $\operatorname{seq}^{1-1}(M)$ denotes the set of all finite sequences without repetition which can be formed with elements of $M$. Furthermore, for a set $S$, let $|S|$ denote the cardinality of $S$. Under the assumption that the four cardinalities $|[M]^2|$, $|M^2|$, $|\operatorname{fin}(M)|$, $|\operatorname{seq}^{1-1}(M)|$ are pairwise distinct and pairwise comparable in ZF, there are six possible linear orderings between these four cardinalities. We show that at least five of the six possible linear orderings are consistent with ZF., Comment: 19 pages, 1 figure

Jing Zhang proved the consistency of $\binom{\omega_2}{\omega_1}\rightarrow\binom{n}{\omega_1}_\omega$ for every $n\in\omega$ with the negative relation $\binom{\omega_2}{\omega_1}\nrightarrow\binom{\omega}{\omega_1}_\omega$. We reduce the consistency strength of this statement to an $\omega_1$-Erdos cardinal.

We look for partition theorems for large subtrees for suitable uncountable trees and colourings. We concentrate on sub-trees of $^{\kappa \ge} 2$ expanded by a well ordering of each level. Unlike earlier works, we do not ask the embedding to preserve the height of the tree but the equality of levels is preserved. We get consistency results without large cardinals. The intention is to apply it to model theoretic problems., Comment: light proof-reading, 16 pages