Topological density of ccc Boolean algebras--Every cardinality occurs

For every uncountable cardinal there is a ccc Boolean algebra whose topological density is .

     

On tightness and depth in superatomic Boolean algebras

We introduce a large cardinal property which is consistent with and show that for every superatomic Boolean algebra and every cardinal with the large cardinal property, if tightness, then depth. This improves a theorem of Dow and Monk.

     

Differentially closed fields

We prove that even the prime, differentially closed field of characteristic zero, is not minimal; that over every differential radical field of characteristicp, there is a closed prime one, and that the theory of closed differential radical fields is stable.     

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.     

Evasion and prediction

 Say that a function π:n ^<ω →n (henceforth called a predictor) k-constantly predicts a real xn ^ω if for almost all intervals I of length k, there is iI such that x(i)=π(x↾i). We study the k-constant prediction number v _n ^const (k), that is, the size of the least family of predictors needed to k-constantly predict all reals, for different values of n and k, and investigate their relationship. Received: 27 June 2001 / Revised version: 10 September 2001 / Published online: 10 October 2002 RID="*" ID="*" Supported by Grant–in–Aid for Scientific Research (C)(2)12640124, Japan Society for the Promotion of Science RID="†" ID="†" Supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Publication 762     

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