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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 721 records · Page 40Linked to original sources
arXiv1996

DOP and FCP in generic structures

Spencer and Shelah [ShSp:304] constructed for each irrational alpha between 0 and 1 the theory T^alpha as the almost sure theory of random graphs with edge probability n^{- alpha}. In [BlSh:528] we proved that this was the same theory as the theory T_alpha built by constructing a generic model in Baldwin and Shi. In this paper we explore some of the more subtle model theoretic properties of this theory. We show that T^alpha has the dimensional order property and does not have the finite cover property.

math.LO↗
arXiv1996

Simple forcing notions and forcing axioms

In the present paper we are interested in simple forcing notions and Forcing Axioms. A starting point for our investigations was the article [JR1] in which several problems were posed. We answer some of those problems here.

math.LO↗
arXiv1996

Existence of almost free abelian groups and reflection of stationary set

section 2: We answer a question of Mekler Eklof on the closure operations of the incompactness spectrum. We answer a question of Foreman and Magidor on reflection of stationary subsets of S_{< aleph_2}(lambda) = {a subseteq lambda : |a| < aleph_2}]. section 3 - NPT is not transitive. We prove NPT(lambda, mu) + NPT(mu, kappa) not => NPT(lambda, kappa)

math.LO↗
arXiv1996

Cardinal preserving ideals

We give some general criteria, when kappa-complete forcing preserves largeness properties -- like kappa-presaturation of normal ideals on lambda (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength ``NS_lambda is aleph_1-preserving'', for lambda > aleph_2 .

math.LO↗
arXiv1996

More Constructions for Boolean algebras

We address a number of problems on Boolean Algebras. For example, we construct, in ZFC, for any BA B, and cardinal kappa BAs B_1,B_2 extending B such that the depth of the free product of B_1,B_2 over B is strictly larger than the depths of B_1 and of B_2 than kappa. We give a condition (for lambda, mu and theta) which implies that for some BA A_theta there are B_1=B^1_{lambda, mu, theta} and B_2B^2_{lambda, mu, theta} such that Depth (B_t) <= mu and Depth (B_1 oplus_{A_theta} B_1) >= lambda. We then investigate for a fixed A, the existence of such B_1,B_2 giving sufficient and necessary conditions, involving consistency results. Further we prove that e.g. if B is a BA of cardinality lambda, lambda >= mu and lambda, mu are strong limit singular of the same cofinality, then B has a homomorphic image of cardinality mu (and with mu ultrafilters). Next we show that for a BA B, if d(B)^kappa <|B| then ind (B)> kappa or Depth (B) >= log (|B|). Finally we prove that if square_lambda holds and lambda = lambda^{aleph_0} then for some BAs B_n, Depth (B_n) <= lambda but for any uniform ultrafilter D on omega, prod_{n< omega} B_n/D has depth >= lambda^+ .

math.LO↗
arXiv1996

Infinite products of finite simple groups

We classify those sequences $\langle S_{n} \mid n \in \mathbb{N} \rangle$ of finite simple nonabelian groups such that the full product $\prod_{n} S_{n}$ has property (FA).

math.GR↗
arXiv1996

Compactness of Loeb Spaces

In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In section 1 we prove that Loeb spaces are compact under various assumptions, and in section 2 we prove that Loeb spaces are not compact under various other assumptions. The results in section 1 and section 2 give a quite complete answer to a question of D. Ross.

math.LO↗
arXiv1996

Cofinalities of elementary substructures of structures on aleph_omega

Let 0 n^*+1 be a function where X subseteq omega backslash (n^*+1) is infinite. Consider the following set S_f= {x subset aleph_omega : |x| <= aleph_{n^*} & (for all n in X)cf(x cap alpha_n)= aleph_{f(n)}}. The question, first posed by Baumgartner, is whether S_f is stationary in [alpha_omega]^{< aleph_{n^*+1}}. By a standard result, the above question can also be rephrased as certain transfer property. Namely, S_f is stationary iff for any structure A=< aleph_omega, ... > there's a B prec A such that |B|= aleph_{n^*} and for all n in X we have cf(B cap aleph_n)= aleph_{f(n)}. In this paper, we are going to prove a few results concerning the above question.

math.LO↗
arXiv1996

Strong covering without squares

We continue [Sh:b, Ch XIII] and [Sh:410]. Let W be an inner model of ZFC. Let kappa be a cardinal in V. We say that kappa-covering holds between V and W iff for all X in V with X subseteq ON and V models |X|< kappa, there exists Y in W such that X subseteq Y subseteq ON and V models |Y|< kappa. Strong kappa-covering holds between V and W iff for every structure M in V for some countable first-order language whose underlying set is some ordinal lambda, and every X in V with X subseteq lambda and V models |X|< kappa, there is Y in W such that X subseteq Y prec M and V models |Y|< kappa. We prove that if kappa is V-regular, kappa^+_V= kappa^+_W, and we have both kappa-covering and kappa^+-covering between W and V, then strong kappa-covering holds. Next we show that we can drop the assumption of kappa^+-covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that kappa^+_W = kappa^+_V and weaken the kappa^+-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).

math.LO↗
arXiv1996

k --Universal Finite Graphs

This paper investigates the class of k-universal finite graphs, a local analog of the class of universal graphs, which arises naturally in the study of finite variable logics. The main results of the paper, which are due to Shelah, establish that the class of k-universal graphs is not definable by an infinite disjunction of first-order existential sentences with a finite number of variables and that there exist k-universal graphs with no k-extendible induced subgraphs.

math.LO↗
arXiv1996

On densities of box products

We construct two universes V_1, V_2 satisfying the following: GCH below \aleph_ω, 2^{\aleph_ω} = \aleph_{ω+2} and the topological density of the space 2^{\aleph_ω} with \aleph_0 box product topology is \aleph_{ω+1} in V_1 and \aleph_{ω+2} in V_2. Further related results are discussed as well.

math.LO↗
arXiv1996

The consistency of 2^{aleph_{0}}> aleph_{omega} + I(aleph_{2})=I(aleph_{omega})

An omega-coloring is a pair where f:[B]^{2} ---> omega. The set B is the field of f and denoted Fld(f). Let f,g be omega-colorings. We say that f realizes the coloring g if there is a one-one function k:Fld(g) ---> Fld(f) such that for all {x,y}, {u,v} in dom(g) we have f({k(x),k(y)}) not= f({k(u),k(v)}) => g({x,y}) not= g({u,v}). We write f~g if f realizes g and g realizes f. We call the ~-classes of omega-colorings with finite fields identities. We say that an identity I is of size r if |Fld(f)|=r for some/all f in I. For a cardinal kappa and f:[kappa]^2 ---> omega we define I(f) to be the collection of identities realized by f and I (kappa) to be bigcap {I(f)| f:[kappa]^2 ---> omega}. We show that, if ZFC is consistent then ZFC + 2^{aleph_0}> aleph_omega + I(aleph_2)=I(aleph_omega) is consistent.

math.LO↗
arXiv1996

On Monk's questions

Monk asks (problems 13, 15 in his list; pi is the algebraic density):''For a Boolean algebra B, aleph_0 <= theta <= pi (B), does B have a subalgebra B' with pi (B')= theta ?'' If theta is regular the answer is easily positive, we show that in general it may be negative, but for quite many singular cardinals - it is positive; the theorems are quite complementary. Next we deal with pi-chi and we show that the pi-chi of an ultraproduct of Boolean algebras is not necessarily the ultraproduct of the pi-chi 's. We also prove that for infinite Boolean algebras A_i (i< kappa) and a non-principal ultrafilter D on kappa : if n_i< aleph_0 for i< kappa and mu = prod_{i< kappa} n_i/D is regular, then pi-chi(A) >= mu. Here A= prod_{i< kappa}A_i/D. By a theorem of Peterson the regularity of mu is needed.

math.LO↗
arXiv1995

Menas' result is best possible

Generalizing some earlier techniques due to the second author, we show that Menas' theorem which states that the least cardinal kappa which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not 2^kappa supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpublished theorem due to the first author.

math.LO↗