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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 757 records · Page 42Linked to original sources
arXiv1995

A game on partial orderings

We study the determinacy of the game G_kappa (A) introduced in [FKSh:549] for uncountable regular kappa and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that G_kappa (A) is undetermined. For the class of linear orders, the existence of such A depends on the size of kappa^{< kappa}. In particular we obtain a characterization of kappa^{< kappa}= kappa in terms of determinacy of the game G_kappa (L) for linear orders L .

math.LO↗
arXiv1995

Identities on cardinals less than aleph_omega

Let kappa be an uncountable cardinal and the edges of a complete graph with kappa vertices be colored with aleph_0 colors. For kappa >2^{aleph_0} the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if kappa <= 2^{aleph_0}, then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when kappa <= 2^{aleph_0}. In particular, we are concerned with the case aleph_1 <= kappa <= aleph_omega

math.LO↗
arXiv1995

Coloring finite subsets of uncountable sets

It is consistent for every (1 <= n< omega) that (2^omega = omega_n) and there is a function (F:[omega_n]^{< omega}-> omega) such that every finite set can be written at most (2^n-1) ways as the union of two distinct monocolored sets. If GCH holds, for every such coloring there is a finite set that can be written at least (sum^n_{i=1}{n+i choose n}{n choose i}) ways as the union of two sets with the same color.

math.LO↗
arXiv1995

Can a small forcing create Kurepa trees?

In the paper we probe the possibilities of creating a Kurepa tree in a generic extension of a model of CH plus no Kurepa trees by an omega_1-preserving forcing notion of size at most omega_1. In the first section we show that in the Levy model obtained by collapsing all cardinals between omega_1 and a strongly inaccessible cardinal by forcing with a countable support Levy collapsing order many omega_1-preserving forcing notions of size at most omega_1 including all omega-proper forcing notions and some proper but not omega-proper forcing notions of size at most omega_1 do not create Kurepa trees. In the second section we construct a model of CH plus no Kurepa trees, in which there is an omega-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions.

math.LO↗
arXiv1995

A consistency result on weak reflection

In this paper we study the notion of strong non-reflection, and its contrapositive weak reflection. We say theta strongly non-reflects at lambda iff there is a function F: theta ---> lambda such that for all alpha < theta with cf(alpha)= lambda there is C club in alpha such that F restriction C is strictly increasing. We prove that it is consistent to have a cardinal theta such that strong non-reflection and weak reflection each hold on an unbounded set of cardinals less than theta .

math.LO↗
arXiv1995

If there is an exactly lambda-free abelian group then there is an exactly lambda-separable one

We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [EM],(p.453). There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly lambda-free ones was proved earlier by the criteria in [Sh:161] in [MkSh:251]. We can apply a similar proof to a large class of other varieties in particular to the variety of (non-commutative) groups.

math.LO↗
arXiv1995

Less nonstationary ideals

We are proving the following: (1) If $\kap$ is a weakly inaccessible then $NS_\kap$ is not $\kap^+$-saturated. (2) If $\kap$ is a weakly inaccessible and $\tet <\kap$ is regular then $NS^\tet_\kap$ is not $\kap^+$-saturated. (3) If $\kap$ is singular then $NS^{cf\kap}_{\kap^+}$ is not $\kap^{++}$-saturated. Combining this with previous results of Shelah, one obtains the following: (A) If $\kap >\aleph_1$ then $NS_\kap$ is not $\kap^+$-saturated. (B) If $\tet^+<\kap$ then $NS^\tet_\kap$ is not $\kap^+$-saturated.

math.LO↗
arXiv1995

Embeddings of Cohen algebras

Complete Boolean algebras proved to be an important tool in topology and set theory. Two of the most prominent examples are B(kappa), the algebra of Borel sets modulo measure zero ideal in the generalized Cantor space {0,1}^kappa equipped with product measure, and C(kappa), the algebra of regular open sets in the space {0,1}^kappa, for kappa an infinite cardinal. C(kappa) is much easier to analyse than B(kappa) : C(kappa) has a dense subset of size kappa, while the density of B(kappa) depends on the cardinal characteristics of the real line; and the definition of C(kappa) is simpler. Indeed, C(kappa) seems to have the simplest definition among all algebras of its size. In the Main Theorem of this paper we show that in a certain precise sense, C(aleph_1) has the simplest structure among all algebras of its size, too. MAIN THEOREM: If ZFC is consistent then so is ZFC + 2^{aleph_0}= aleph_2 +``for every complete Boolean algebra B of uniform density aleph_1, C(aleph_1) is isomorphic to a complete subalgebra of B''.

math.LO↗
arXiv1995

Abstract classes with few models have `homogeneous-universal' models

This paper is concerned with a class K of models and an abstract notion of submodel <=. Experience in first order model theory has shown the desirability of finding a `monster model' to serve as a universal domain for K. In the original constructions of Jonsson and Fraisse, K was a universal class and ordinary substructure played the role of <=. Working with a cardinal lambda satisfying lambda^{< lambda}= lambda guarantees appropriate downward Lowenheim-Skolem theorems; the existence and uniqueness of a homogeneous-universal model appears to depend centrally on the amalgamation property. We make this apparent dependence more precise in this paper. The major innovation of this paper is the introduction of weaker notion to replace the natural notion of (K, <=)-homogeneous-universal model. Modulo a weak extension of ZFC (provable if V=L), we show that a class K obeying certain minimal restrictions satisfies a fundamental dichotomy: For arbitrarily large lambda, either K has the maximal number of models in power lambda or K has a unique chain homogenous-universal model of power lambda. We show that in a class with amalgamation this dichotomy holds for the notion of K-homogeneous-universal model in the more normal sense.

math.LO↗
arXiv1995

On the strong equality between supercompactness and strong compactness

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V models ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V[G] models ZFC + GCH in which, (a) (preservation) for kappa <= lambda regular, if V models ``kappa is lambda supercompact'', then V[G] models ``kappa is lambda supercompact'' and so that, (b) (equivalence) for kappa <= lambda regular, V[G] models ``kappa is lambda strongly compact'' iff V[G] models ``kappa is lambda supercompact'', except possibly if kappa is a measurable limit of cardinals which are lambda supercompact.

math.LO↗
arXiv1995

The pcf-theorem revisited

The pcf theorem (of the possible cofinality theory) was proved for reduced products prod_{i< kappa} lambda_i/I, where kappa < min_{i< kappa} lambda_i. Here we prove this theorem under weaker assumptions such as wsat(I)< min_{i< kappa} lambda_i, where wsat(I) is the minimal theta such that kappa cannot be delivered to theta sets notin I (or even slightly weaker condition). We also look at the existence of exact upper bounds relative to <_I (<_I-eub) as well as cardinalities of reduced products and the cardinals T_D(lambda). Finally we apply this to the problem of the depth of ultraproducts (and reduced products) of Boolean algebras

math.LO↗
arXiv1995

The tree property at successors of singular cardinals

Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.

math.LO↗
arXiv1995

Convergence in homogeneous random graphs

For a sequence p=(p(1),p(2), ...) let G(n,p) denote the random graph with vertex set {1,2, ...,n} in which two vertices i, j are adjacent with probability p(|i-j|), independently for each pair. We study how the convergence of probabilities of first order properties of G(n,p), can be affected by the behaviour of p and the strength of the language we use.

math.LO↗
arXiv1995

The Bounded Proper Forcing Axiom

The bounded proper forcing axiom BPFA is the statement that for any family of aleph_1 many maximal antichains of a proper forcing notion, each of size aleph_1, there is a directed set meeting all these antichains. A regular cardinal kappa is called {Sigma}_1-reflecting, if for any regular cardinal chi, for all formulas phi, ``H(chi) models `phi ' '' implies ``exists delta < kappa, H(delta) models `phi ' '' We show that BPFA is equivalent to the statement that two nonisomorphic models of size aleph_1 cannot be made isomorphic by a proper forcing notion, and we show that the consistency strength of the bounded proper forcing axiom is exactly the existence of a Sigma_1-reflecting cardinal (which is less than the existence of a Mahlo cardinal). We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure.

math.LO↗
arXiv1995

On invariants for omega_1-separable groups

We study the classification of omega_1-separable groups using Ehrenfeucht-Fraisse games and prove a strong classification result assuming PFA, and a strong non-structure theorem assuming diamond.

math.LO↗