Consistently There Is No Non Trivial CCC Forcing Notion with the Sacks or Laver Property

Dedicated to the memory of Paul Erdős Received March 2, 2000 RID=" " ID=" " The research supported by the Israel Science Foundation (founded by the Israel Academy of Sciences). Publication number 723. We would like to thank Martin Goldstern for many improvements in this paper.     

Bousfield localization as an algebraic closure of groups

We show that the notion ofHZ-local groups due to A. Bousfield which is based on considerations from algebraic topology can be defined and understood in terms of solubility of certain systems of equations overG.     

Every null-additive set is meager-additive

It is proved that every null-additive subset of^ω2 is meager-additive. Several characterizations of the null-additive subsets of^ω2 are given, as well as a characterization of the meager additive subsets of^ω2. Under CH, an uncountable null-additive subset of^ω2 is constructed. Publ. No. 445. First version written in April 1991. Partially supported by the Basic Research Fund of the Israel Academy of Sciences and the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Convergence in homogeneous random graphs

For a sequence p? = (p(1), p(2), ‥ .), let G(n, p?) denote the random graph with vertex set {1, 2, ‥ ., n} in which two vertices i, j are adjacent with probability p(|i ? j|), independently for each pair. We study how the convergence of probabilities of first order properties of G(n, p?), can be affected by the behaviour of p? and the strength of the language we use.     

A weak generalization of MA to higher cardinals

We generalized MA e.g., to ℵ₁-complete forcing, by strengthening the ℵ₂-C.C. condition which occurs in many proofs. We show some consequences of MA generalized, and show that we get a model of ZFC in which the modadic theory ofω ₂ is decidable. The author thanks the United States-israel Binational Foundation for partially supporting his research by Grant 1110.     

A proof of vaught’s conjecture forω-stable theories

In this paper it is proved that ifT is a countable completeω-stable theory in ordinary logic, thenT has either continuum many, or at most countably many, non-isomorphic countable models. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author thanks the United States-Israel Binational Science Foundation for supporting his research. Supported by the Natural Sciences and Engineering Research Council of Canada, and FCAC Quebec.     

More on the pressing down game

We investigate the pressing down game and its relation to the Banach Mazur game. In particular we show: consistently, there is a nowhere precipitous normal ideal I on ${\aleph_2}$ such that player nonempty wins the pressing down game of length ${\aleph_1}$ on I even if player empty starts.     

A Hanf number for saturation and omission: the superstable case

Suppose is a triple of two theories in vocabularies with cardinality λ, and a τ₁‐type p over the empty set that is consistent with T₁. We consider the Hanf number for the property “there is a model M₁ of T₁ which omits p, but is saturated”. In [2], we showed that this Hanf number is essentially equal to the Löwenheim number of second order logic. In this paper, we show that if T is superstable, then the Hanf number is less than .