Large weight does not yield an irreducible base

Answering a question of Juhász, Soukup and Szentmiklóssy, we show that it is consistent that some first countable space of uncountable weight does not contain an uncountable subspace which has an irreducible base.     

Existence of EF‐equivalent non‐isomorphic models

We prove the existence of pairs of models of the same cardinality λ which are very equivalent according to EF games, but not isomorphic. We continue the paper [4], but we do not rely on it. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

How special are Cohen and random forcings,i.e. boolean algebras of the family of subsets of reals modulo meagre or null

We prove that any Souslin c.c.c. forcing notion which adds a nondominated real adds a Cohen real. We also prove that any Souslin c.c.c. forcing adds a real which is not on any old “narrow” tree.     

Hereditarily separable groups and monochromatic uniformization

We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality ?₁. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove that it is consistent that every Whitehead group is free but not every hereditarily separable group is free. We also show that the fact that ? is a p.i.d. with infinitely many primes is essential for this result.     

Models of expansions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb N}$\end{document} with no end extensions

We deal with models of Peano arithmetic (specifically with a question of Ali Enayat). The methods are from creature forcing. We find an expansion of ${\mathbb N}$ such that its theory has models with no (elementary) end extensions. In fact there is a Borel uncountable set of subsets of ${\mathbb N}$ such that expanding ${\mathbb N}$ by any uncountably many of them suffice. Also we find arithmetically closed ${\mathcal A}$ with no ultrafilter on it with suitable definability demand (related to being Ramsey). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On strongly-non-reflexive groups

A strong negative answer is given to the old question of whether every dual group is reflexive. Using ◊_ω1 a groupA is constructed so thatA, A*, A**, andA*** are weakly ω₁-separable groups of cardinalityω ₁ andA* is not isomorphic toA***. Research partially supported by NSF Grant No. DMS-8400451. Research partially supported by NSERC Grant No. A8948.     

A model in which there are Jech-Kunen trees but there are no Kurepa trees

By an ω₁-tree we mean a tree of power ω₁ and height ω₁. We call an ω₁-tree a Jech-Kunen tree if it hask-many branches for somek strictly between ω₁ and 2^ω1. In this paper we construct the models of CH plus 2^ω1 > ω₂, in which there are Jech-Kunen trees and there are no Kurepa trees. The research of the first author was partially supported by the Basic Research Fund, Israeli Academy of Science, Publ. No. 466.     

Cofinalities of elementary substructures of structures on ℵω

We give some sufficient conditions for {a ⊆ ℵ_ω: cf(sup(a⊆ℵ _n )) =f(n) for to be stationary. Thus we get consistency e.g. of cases wheref has ≥ three values attained infinitely often. We use pcf, partition theorems on trees and games. Partially supported by the United States-Israel Binational Science Foundation, Publication 484.     

Comparing the uniformity invariants of null sets for different measures

It is shown to be consistent with set theory that the uniformity invariant for Lebesgue measure is strictly greater than the corresponding invariant for Hausdorff r-dimensional measure where 0<r<1.