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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 253 records · Page 14Linked to original sources
arXiv2016

A Framework for Forcing Constructions at Successors of Singular Cardinals

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size is less than $2^κ$ and such that any graph on $κ^+$ embeds into one of the graphs in the collection.

math.LO↗
arXiv2016

Set mappings on 4-tuples

In this paper we study set mappings on 4-tuples. We continue a previous work of Komjath and Shelah by getting new finite bounds on the size of free sets in a generic extension. This is obtained by an entirely different forcing construction. Moreover we prove a ZFC result for set mappings on 4-tuples and also as another application of our forcing construction we give a consistency result for set mappings on triples.

math.LO↗
arXiv2016

There is no bound on sizes of indecomposable Banach spaces

Assuming the generalized continuum hypothesis we construct arbitrarily big indecomposable Banach spaces. i.e., such that whenever they are decomposed as $X\oplus Y$, then one of the closed subspaces $X$ or $Y$ must be finite dimensional. It requires alternative techniques compared to those which were initiated by Gowers and Maurey or Argyros with the coauthors. This is because hereditarily indecomposable Banach spaces always embed into $\ell_\infty$ and so their density and cardinality is bounded by the continuum and because dual Banach spaces of densities bigger than continuum are decomposable by a result due to Heinrich and Mankiewicz. The obtained Banach spaces are of the form $C(K)$ for some compact connected Hausdorff space and have few operators in the sense that every linear bounded operator $T$ on $C(K)$ for every $f\in C(K)$ satisfies $T(f)=gf+S(f)$ where $g\in C(K)$ and $S$ is weakly compact or equivalently strictly singular. In particular, the spaces carry the structure of a Banach algebra and in the complex case even the structure of a $C^*$-algebra.

math.FA↗
arXiv2016

Decidability and classification of the theory of integers with primes

We show that under Dickson's conjecture about the distribution of primes in the natural numbers, the theory Th(Z,+,1,0,Pr) where Pr is a predicate for the prime numbers and their negations is decidable, unstable and supersimple. This is in contrast with Th(Z,+,0,Pr,<) which is known to be undecidable by the works of Jockusch, Bateman and Woods.

math.LO↗
arXiv2016

General Non-structure Theory

The theme of the first two sections, is to prepare the framework of how from a "complicated" family of 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 graph, for this we phrase relevant complicatedness properties (called bigness). We say when M in K_2 is represented in I in K_1. We give sufficient conditions when {M_I:I\in K^1_λ} is complicated where for each I in K^1_lambda we build M_I in K^2 (usually in K^2_lambda) represented in it and reflecting to some degree its structure (e.g. for I a linear order we can build a model of an unstable first order class reflecting the order). If we understand enough we can even build e.g. rigid members of K^2_lambda. Note that we mention "stable", "superstable", but in a self contained way, using an equivalent definition which is useful here and explicitly given. We also frame the use of generalizations of Ramsey and Erdos-Rado theorems to get models in which any I from the relevant K_1 is reflected. We give in some detail how this may apply to the class of separable reduced Abelian p-group and how we get relevant models for ordered graphs (via forcing). 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 so that for enough cuts of I, their being omitted is reflected in M_I, then there are 2^lambda non-isomorphic cases.

math.LO↗
arXiv2016

Compactness of the quantifier on "Complete Embedding of BA's"

We try to build, provably in ZFC, for a first order T a model in which any isomorphism between two Boolean algebras is definable. The problem, compared to [Sh:384], is with pseudo-finite Boolean algebras. A side benefit is that we do not use Skolem function (which do not matter for proving compactness of logics but still are of interest). Let lambda be 2^mu if regular and its successor otherwise. Model theoretically we investigate notions of bigness of types, usually those are ideals of the set of formulas in a model, definable in appropriate sense. We build a model of cardinality lambda^plus by a sequence of models M_alpha of cardinality lambda for alpha less than lambda^plus, each M_alpha equips with a sequence (M_alpha, i, a_alpha, i, Omega_alpha, i) : i in S_I subseteq lambda, with M_alpha, i is of cardinality less than lambda, precedes-increasing continuous with i, Omega_alpha, i a bigness notion defined using parameters from M_alpha, i and a_alpha, i realized in M_alpha, i plus 1 over M_alpha,i a Omega_alpha, i-big type. As alpha increase, not only M_alpha increase, but this extra structure increasing modulo a club of lambda, this is why we have insisted on lambda being regular. This can be considered as a way to omit types of cardinality lambda, which in general is hard. The fact that lambda is not too much larger than mu helps us to guarantee that any possible automorphism of structures be defined in M equals union bracket M_alpha: alpha less than lambda^plus bracket by approximations of cardinal mu and so we can enumerate them all.

math.LO↗
arXiv2016

On the spectrum of characters of ultrafilters

We show that the character spectrum $Sp_χ(λ)$, for a singular cardinal $λ$ of countable cofinality, may include any prescribed set of regular cardinals between $λ$ and $2^λ$.

math.LO↗
arXiv2015

Combinatorial background for non-structure

This was supposed to be an appendix to the book "Non-structure", and probably will be if it materializes. It presents relevant material, sometimes new, which was used in works which were supposed to be part of that book. In section 1 we deal with partition theorems on trees with omega levels; it is self contained. In section 2 we deal with linear orders which are countable union of scattered ones with unary predicated, it is self contained. In section 3 we deal mainly with pcf theory but just quote. In section 4, on normal ideals, we repeat [Sh:247]. This is used in [Sh:331].

math.LO↗
arXiv2015

Random graphs and Lindstrom quantifiers for natural graph properties

We study zero-one laws for random graphs. We focus on the following question that was asked by many: Given a graph property P, is there a language of graphs able to express P while obeying the zero-one law? Our results show that on the one hand there is a (regular) language able to express connectivity and k-colorability for any constant k and still obey the zero-one law. On the other hand we show that in any (semiregular) language strong enough to express Hamiltonicity one can interpret arithmetic and thus the zero-one law fails miserably. This answers a question of Blass and Harary.

math.LO↗
arXiv2015

Exact saturation in simple and NIP theories

A theory $T$ is said to have exact saturation at a singular cardinal $κ$ if it has a $κ$-saturated model which is not $κ^{+}$-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.

math.LO↗
arXiv2015

Creature forcing and five cardinal characteristics in Cichoń's diagram

We use a (countable support) creature construction to show that consistently \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{non}(\text{NULL}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \] The same method shows the consistency of \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \]

math.LO↗
arXiv2015

On non-forking spectra

Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point of a variety). To a countable first-order theory we associate its non-forking spectrum - a function of two cardinals kappa and lambda giving the supremum of the possible number of types over a model of size lambda that do not fork over a sub-model of size kappa. This is a natural generalization of the stability function of a theory. We make progress towards classifying the non-forking spectra. On the one hand, we show that the possible values a non-forking spectrum may take are quite limited. On the other hand, we develop a general technique for constructing theories with a prescribed non-forking spectrum, thus giving a number of examples. In particular, we answer negatively a question of Adler whether NIP is equivalent to bounded non-forking. In addition, we answer a question of Keisler regarding the number of cuts a linear order may have. Namely, we show that it is possible that ded(kappa) < ded(kappa)^omega.

math.LO↗