On the Number of Elementary Submodels of an Unsuperstable Homogeneous Structure

We show that if M is a stable unsuperstable homogeneous structure, then for most κ ? |M|, the number of elementary submodels of M of power κ is 2^κ.     

Aut (M) Has a Large Dense Free Subgroup for Saturated M

We prove that for a stable theory T, if M is a saturated modelof T cardinality where > |T|, then Aut (M) has a densefree subgroup on 2 generators. This affirms a conjecture ofHodges.     

Universal abelian groups

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example:Theorem:For n≧2, there is a purely universal separable p-group in ℵ _n if, and only if, . Partially supported by the United States-Israel Binational Science Foundation. Publication number 455.     

Stable structures homogeneous for a finite binary language

LetL be a finite relational language andH(L) denote the class of all countable stableL-structuresM for which Th(M) admits elimination of quantifiers. ForM ∈H(L) define the rank ofM to be the maximum value of CR(p, 2), wherep is a complete 1-type over Ø and CR(p, 2) is Shelah’s complete rank. IfL has only unary and binary relation symbols there is a uniform finite bound for the rank ofM ∈H(L). This theorem confirms part of a conjecture of the first author. Intuitively it says that for eachL there is a finite bound on the complexity of the structures inH(L).     

The structure of saturated free algebras

Requiring an algebraM to be both free (for the variety it generates) and ₁-saturated imposes very strong conditions onM. In the simplest examples (see below) there exist a finite number of relatively free algebrasA _o,...,A _n-1 whose theories are ₁-categorical such thatM is generated (as an algebra) by the UA _i In particular, this implies Th(M) has at most (+_o) models of cardinality . We will show a weaker structure theorem in the general case but deduce the same constraint on the spectrum ofT.Dedicated to Alfred Tarski on his 80th birthdayPartially supported by N.S.F. grant 77-01667.This research was partially supported by the United States Israel Binational Science Foundation grant 1110.Presented by R. McKenzie.     

S-forcing IIa: Adding diamonds and more applications: Coding sets,Arhangel’skii’s problem and ℒ[Q_1^{< \omega } ,Q_2^1 ]

We continue our earlier paper [20] by proving the equivalence, for regularκ>ω, of the existence of (κ, 1) morasses with built-in ♦ sequences and a strengthening, S_K◊ , of the forcing principle, S_K◊ of [20]. We obtain various applications of S_K◊, to wit: the existence of a stationary subset of [K₊]_<K with sup as coding function, the existence of a counterexample to Arhangel’skii’s conjecture (κ=ℵ₁) and compactness, axiomatizability and transfer properties for the Magidor-Malitz language ℒ (κ=ℵ₁). Research partially supported by NSF Grant MCS 8301042.     

There are no very meager sets in the model in which both the Borel Conjecture and the dual Borel Conjecture are true

We show that the model for the simultaneous consistency of the Borel Conjecture and the dual Borel Conjecture given in  4 actually satisfies a stronger version of the dual Borel Conjecture: there are no uncountable very meager sets.     

Stable theories and representation over sets

In this paper we give characterizations of the stable and ?₀‐stable theories, in terms of an external property called representation. In the sense of the representation property, the mentioned classes of first‐order theories can be regarded as “not very complicated”.     

EF-equivalent not isomorphic pair of models

We construct non-isomorphic models , e.g. of cardinality , such that in the Ehrenfeucht-Fraissé game of any length the isomorphism player wins.

     

Refuting ehrenfeucht conjecture on rigid models

We prove that the class of cardinalities in which a first-order sentence has a rigid model can be very complicated, and we essentially characterize the possible classes. Dedicated to the memory of A. Robinson.