A tall space with a small bottom

We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if , then there is such a space of height with only many isolated points. This implies that there is a locally compact scattered space of height with isolated points in ZFC, solving an old problem of the first author.

     

Not collapsing cardinals ≤<Emphasis Type="Italic">κ</Emphasis> in (<<Emphasis Type="Italic">κ</Emphasis>)-support iterations

We deal with the problem of preserving various versions of completeness in (<κ)-support iterations of forcing notions, generalizing the case “S-complete proper is preserved by CS iterations for a stationary costationaryS⊆ω ₁”. We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinalκ and a stationary setS⊆κ with fat complement we can have uniformization for every (A _δ :δ ∈S′),A _δ ⊆δ = supA _δ , cf (δ) = otp(A _δ ) and a stationary non-reflecting setS′⊆S (see B.8.2). Research supported by The German-Israeli Foundation for Scientific Research & Development Grant No. G-294.081.06/93 and by The National Science Foundation Grant No. 144-EF67. Publication No. 587.     

Strongly meager sets of size continuum

We will construct several models where there are no strongly meager sets of size 2⁰. First author partially supported by NSF grant DMS 0200671.Second author partially supported by Israel Science Foundation and NSF grant DMS 0072560. Publication 807. Mathematics Subject Classification (2000):03E15, 03E20     

On ultraproducts of Boolean algebras and irr

1. Consistent inequality [We prove the consistency of irrirr(B_i)/D where D is an ultrafilter on and each B_i is a Boolean algebra and irr(B) is the maximal size of irredundant subsets of a Boolean algebra B, see full definition in the text. This solves the last problem, 35, of this form from Monk's list of problems in [M2]. The solution applies to many other properties, e.g. Souslinity.] 2. Consistency for small cardinals [We get similar results with =₁ (easily we cannot have it for =₀) and Boolean algebras B_i (i<) of cardinality .] This article continues Magidor Shelah [MgSh:433] and Shelah Spinas [ShSi:677], but does not rely on them: see [M2] for the background. I would like to thank Alice Leonhardt for the beautiful typing. This research was partially supported by the Israel Science Foundation. Publication 703     

On the number of -equivalent non-isomorphic models

We prove that if is consistent then is consistent with the following statement: There is for every a model of cardinality which is -equivalent to exactly non-isomorphic models of cardinality . In order to get this result we introduce ladder systems and colourings different from the ``standard' counterparts, and prove the following purely combinatorial result: For each prime number and positive integer it is consistent with that there is a ``good' ladder system having exactly pairwise nonequivalent colourings.

     

Simple complete Boolean algebras

For every regular cardinal there exists a simple complete Boolean algebra with generators.

     

On reflection of stationary sets in

Let be an inaccessible cardinal, and let and is regular and . It is consistent that the set is stationary and that every stationary subset of reflects at almost every .

     

Absolutely indecomposable modules

A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more general result about -modules over a large class of commutative rings with endomorphism ring which remains the same when passing to a generic extension of the universe. It turns out that `large' in this context has a precise meaning, namely being smaller than the first -Erdos cardinal defined below. We will first apply a result on large rigid valuated trees with a similar property established by Shelah in 1982, and will prove the existence of related `-modules' (-modules with countably many distinguished submodules) and finally pass to -modules. The passage through -modules has the great advantage that the proofs become very transparent essentially using a few `linear algebra' arguments also accessible for graduate students. The result closes a gap of Eklof and Shelah (1999) and Eklof and Mekler (2002), provides a good starting point for Fuchs and Göbel, and gives a new construction of indecomposable modules in general using a counting argument.

     

On certain indestructibility of strong cardinals and a question of Hajnal

A model in which strongness of is indestructible under ⁺ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of { |2 <}.     

Random graphs in the monadic theory of order

We continue the works of Gurevich-Shelah and Lifsches-Shelah by showing that it is consistent with ZFC that the first-order theory of random graphs is not interpretable in the monadic theory of all chains. It is provable from ZFC that the theory of random graphs is not interpretable in the monadic second order theory of short chains (hence, in the monadic theory of the real line). Received: 18 July 1996