On topological properties of ultraproducts of finite sets

In [3] a certain family of topological spaces was introduced on ultraproducts. These spaces have been called ultratopologies and their definition was motivated by model theory of higher order logics. Ultratopologies provide a natural extra topological structure for ultraproducts. Using this extra structure in [3] some preservation and characterization theorems were obtained for higher order logics. The purely topological properties of ultratopologies seem interesting on their own right. We started to study these properties in [2], where some questions remained open. Here we present the solutions of two such problems. More concretely we show 1. that there are sequences of finite sets of pairwise different cardinalities such that in their certain ultraproducts there are homeomorphic ultratopologies and 2. if A is an infinite ultraproduct of finite sets, then every ultratopology on A contains a dense subset D such that |D| < |A|. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generating ultrafilters in a reasonable way

We continue investigations of reasonable ultrafilters on uncountable cardinals defined in Shelah [8]. We introduce a general scheme of generating a filter on λ from filters on smaller sets and we investigate the combinatorics of objects obtained this way. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cofinality of the nonstationary ideal

We show that the reduced cofinality of the nonstationary ideal on a regular uncountable cardinal may be less than its cofinality, where the reduced cofinality of is the least cardinality of any family of nonstationary subsets of such that every nonstationary subset of can be covered by less than many members of . For this we investigate connections of the various cofinalities of with other cardinal characteristics of and we also give a property of forcing notions (called manageability) which is preserved in -support iterations and which implies that the forcing notion preserves non-meagerness of subsets of (and does not collapse cardinals nor changes cofinalities).

     

Cotorsion theories and splitters

Let be a subring of the rationals. We want to investigate self splitting -modules (that is . Following Schultz, we call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples of splitters. Are there others? Answering an open problem posed by Schultz, we will show that there are more splitters, in fact we are able to prescribe their endomorphism -algebras with a free -module structure. As a by-product we are able to solve a problem of Salce, showing that all rational cotorsion theories have enough injectives and enough projectives. This is also basic for answering the flat-cover-conjecture.

     

The number of non-isomorphic models of an unstable first-order theory

It is proved that ifT is an unstable (first-order) theory,λ>|T|+ℵ₀, thenT has exactly 2 ^λ non-isomorphic models of cardinalityλ. In fact we have stronger results: this is true for pseudo-elementary classes, and for almost everyλ≧|T|+ℵ₁. The preparation of this paper was sponsored in part by NSF Grant GP-22937. This work was supported in part by NSF Grant GP-22794.     

Models with second order properties IV. A general method and eliminating diamonds

We show how to build various models of first-order theories, which also have properties like: tree with only definable branches, atomic Boolean algebras or ordered fields with only definable automorphisms.For this we use a set-theoretic assertion, which may be interesting by itself on the existence of quite generic subsets of suitable partial orders of power λ⁺, which follows from ?_λ and even weaker hypotheses (e.g., λ=?₀, or λ strongly inaccessible). For a related assertion, which is equivalent to the morass see Shelah and Stanley [16].The various specific constructions serve also as examples of how to use this set-theoretic lemma. We apply the method to construct rigid ordered fields, rigid atomic Boolean algebras, trees with only definable branches; all in successors of regular cardinals under appropriate set- theoretic assumptions. So we are able to answer (under suitable set-theoretic assumptions) the following algebraic question.Saltzman's Question. Is there a rigid real closed field, which is not a subfield of the reals?     

Classification theory for non-elementary classes I: The number of uncountable models of\psi \in L_{w_1 ,w} . Part B. Part B

We continue here Part A, and the main results are proved here. This part deals withn-dimensional diagrams of models.     

Decomposing uncountable squares to countably many chains

We construct an ordered set I of cardinality $\text{N}$₁, such that its square is the union of $\text{N}$₀ chains (in the natural partial order).     

On languages with non-homogeneous strings of quantifiers

It is shown that every linear string of quantifiers can be replaced by a well-ordered sequence of quantifiers. The results of this paper was part of the author’s Master Thesis, which was submitted to the Hebrew University in August, 1967. The work was done under the guidance of Professor H. Gaifman, whom I thank for his kind guidance, and his help to clarify the exposition in this article.