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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 109 records · Page 6Linked to original sources
arXiv2022

On projections of the tails of a power

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

math.LO↗
arXiv2022

The Keisler-Shelah isomorphism theorem and the continuum hypothesis II

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 $\mathbf m=\langle (\mathbb{M}^{1}_n, \mathbb{M}^{2}_n: n < ω\rangle$ 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 $ω$, 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.

math.LO↗
arXiv2022

Ramsey partitions of metric spaces

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~\(\aleph_1\).

math.LO↗
arXiv2022

Non-existence of universals for classes like reduced torsion free abelian groups under embeddings which are not necessarily pure

We consider a class K of structures e.g. trees with omega +1 levels, metric spaces and mainly, classes of Abelian groups like the one mentioned in the title and the class of reduced separable (Abelian) p-groups. We say M in K is universal for K if any member N of K of cardinality not bigger than the cardinality of M can be embedded into M . This is a natural, often raised, problem. We try to draw consequences of cardinal arithmetic to non-existence of universal members for such natural classes.

math.LO↗
arXiv2022

Analytical Guide and updates for "Cardinal Arithmetic"

Part A: A revised version of the guide in "Cardinal Arithmetic" ([Sh:g]), with corrections and expanded to include later works. Part B: Corrections to [Sh:g]. Part C: Contains some revised proof and improved theorems. Part D: Contains a list of relevant references.

math.LO↗
arXiv2022

On Spectrum of kappa-Resplendent Models

We prove that some natural "outside" property is equivalent (for a first order class) to being stable. For a model, being resplendent is a strengthening of being kappa-saturated. Restricting ourselves to the case kappa > |T| for transparency, a model M is kappa-resplendent means: when we expand M by <kappa individual constants (c_i:i < alpha), if (M, c_i)_ {i<alpha} has an elementary extension expandable to be a model of T' where Th((M,c_i)_{i<alpha}) subseteq T', |T'| < kappa then already (M,c_i)_{i<α} can be expanded to a model of T'. Trivially any saturated model of cardinality lambda is lambda-resplendent. We ask: how may kappa-resplendent models of a (first order complete) theory T of cardinality lambda are there? Naturally we restrict ourselves to cardinals lambda=lambda^ kappa + 2^ {|T|}. Then we get a complete and satisfying answer: this depends just on T being stable or unstable. In this case proving that for stable T we get few, is not hard; in fact every resplendent model of T is saturated hence determined by its cardinality up to isomorphism. The inverse is more problematic because naturally we have to use Skolem functions with any alpha<kappa places. Normally we use relevant partition theorems (Ramsey theorem or Erdős-Rado theorem), but in our case the relevant partitions theorems fails so we have to be careful.

math.LO↗
arXiv2022

Nice aleph_1 generated non-P-points, I

We define a family of a (non-principal) ultrafilters on N which are, in a sense, far from P-points. We first under reasonable conditions, prove its existence. In a continuation we shall prove that such a point may exist while no P-point exists.

math.LO↗
arXiv2022

On CON(Dominating_lambda > cov_λ(meagre))

We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e. the cofinality of lambda^lambda is strictly bigger than cov(meagre_lambda), i.e. the minimal number of nowhere dense subsets of 2^lambda needed to cover it. This answers a question of Matet.

math.LO↗
arXiv2022

Corrected Iteration

For $λ$ inaccessible, we may consider $(< λ)$-support iteration of some specific $(<λ)$-complete $λ^+$-c.c. forcing notion. But this fails a "preservation by restricting to a sub-sequence of the forcing, we "correct" the iteration to regain it. This is used in another paper in the consistency of $cov(meagre) < \mathfrak{d}_λ$.

math.LO↗
arXiv2022

A collection of abstracts of Shelah's Papers

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).

math.LO↗
arXiv2022

Between reduced powers and ultrapowers, II

We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fréchet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.

math.LO↗
arXiv2022

S-spaces and large continuum

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.

math.LO↗
arXiv2022

Graphs represented by Ext

This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph $(μ,R)$ we can find a family $\{\mathbb G_α: α< μ\}$ of abelian groups such that for each $α,β\inμ$: $$Ext(\mathbb G_α, \mathbb G_β) = 0 \Longleftrightarrow(α,β) \in R.$$ 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 $λ= λ^{\aleph_0}$ and $μ= 2^λ$. We apply Jensen's diamond principle along with the criteria to present $λ$-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.

math.AC↗
arXiv2022

NNR Revisited

We show if we use countable support iteration of forcing notions not adding reals that satisfy additional conditions, then the limit forcing does not add reals. As a result we prove that we can amalgamate two earlier methods and prove the consistency with ZFC + GCH of two statements gotten separately earlier: Souslin hypothesis and non-club guessing. We also answer a question of Justin Moore by proving the consistency of one further case of "strong failure of club guessing" with GCH.

math.LO↗
arXiv2022

The Keisler-Shelah isomorphism theorem and the continuum hypothesis

We show that if for any two elementary equivalent structures $\mathbf{M}, \mathbf{N}$ of size at most continuum in a countable language, $\mathbf{M}^ω/ \mathcal{U} \simeq \mathbf{N}^ω/ \mathcal{U}$ for some ultrafilter $\mathcal{U}$ on $ω,$ then $CH$ holds. We also provide some consistency results about Keisler and Shelah isomorphism theorems in the absence of $CH$.

math.LO↗
arXiv2022

Exact saturation in pseudo-elementary classes for simple and stable theories

We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.

math.LO↗
arXiv2022

Borel sets without perfectly many overlapping translations, II

For a countable ordinal epsilon we construct a Sigma^0_2 subset of the Cantor space for which one may force aleph_epsilon translations with intersections of size 2i, but such that it has no perfect set of such translations in any ccc extension. These sets have uncountably many translations with intersections of size 2i in ZFC, so this answers Problem 3.4 of arxiv:1711.04058 .

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