The theorems of Beth and craig in abstract model theory,III: Δ-logics and infinitary logics

We give a general technique on how to produce counterexamples to Beth's definability (and weak definability) theorem. The method is then applied for various infinitary, cardinality quantifier logics and Δ-closure of such logics.     

Cofinality of normal ideals on<Emphasis Type="Italic">P</Emphasis><Subscript>κ</Subscript>(λ) II

For an idealJ on an infinite setX with add(J)=κ, let be the smallest size of any subfamilyY ofJ with the property that any member ofJ can be covered by less than κ members ofY. We study the value of forA in , where denotes the smallest [δ]^<θ ideal onP _κ(λ). We also discuss the problem of whether there exists a setA such that , or even . Some of the material in this paper originally appeared as part of the author's doctoral dissertation completed at the Université de Caen, 1998. Partially supported by the Israel Science Foundation. Publication 813.     

A note on extensions of infinitary logic

We show that a strong form of the so called Lindströms Theorem [4] fails to generalize to extensions of L and L : For weakly compact there is no strongest extension of L with the (,)-compactness property and the Löwenheim-Skolem theorem down to . With an additional set-theoretic assumption, there is no strongest extension of L with the (,)-compactness property and the Löwenheim-Skolem theorem down to <.We are indebted to Lauri Hella, Tapani Hyttinen and Kerkko Luosto for useful suggestions.Research partially supported by the United States-Israel Binational Science Foundation. Publication number [ShVa:726]Research partially supported by grant 40734 of the Academy of Finland.     

A closed (n + 1)-convex set inR 2 is a union ofn 6 convex sets

It is shown that if a closed setS in the plane is (n+1)-convex, then it has no more thann ⁴ holes. As a consequence, it can be covered by≤n ⁶ convex subsets. This is an improvement on the known bound of 2 ⁿ ·n ³. The author would like to thank the BSF for partially supporting this research. Publication no. 354.     

Weak Reflection at the Successor of a Singular Cardinal

The notion of stationary reflection is one of the most importantnotions of combinatorial set theory. Weak reflection, whichis, as its name suggests, a weak version of stationary reflection,is investigated. The main result is that modulo a large cardinalassumption close to 2-hugeness, there can be a regular cardinal such that the first cardinal weakly reflecting at is the successorof a singular cardinal. This answers a question of Cummings,Damonja and Shelah.     

Specialising Aronszajn trees by countable approximations

We show that there are proper forcings based upon countable trees of creatures that specialise a given Aronszajn tree.The first author was partially supported by a Minerva fellowship. The second author's research was partially supported by the ``Israel Science Foundation', founded by the Israel Academy of Science and Humanities. This is the second author's work number 778. Mathematics Subject Classification (2000): 03E15, 03E17, 03E35, 03D65     

Van der Waerden spaces and Hindman spaces are not the same

A Hausdorff topological space is van der Waerden if for every sequence in there is a converging subsequence where contains arithmetic progressions of all finite lengths. A Hausdorff topological space is Hindman if for every sequence in there is an IP-converging subsequence for some infinite .

We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.

     

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.