Related papers: Filtration equivalent aleph_1-separable abelian gr…

We show that it is consistent that there is a strongly aleph_1-free aleph_1-coseparable group of cardinality aleph_1 which is not aleph_1-separable.

We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for aleph_1-separable abelian groups of cardinality aleph_1. In fact, there is an aleph_1-separable…

The connections between Whitehead groups and uniformization properties were investigated by the third author in [Sh:98]. In particular it was essentially shown there that there is a non-free Whitehead (respectively, aleph_1-coseparable)…

We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality aleph_1. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove…

For a given stationary set $S$ of countable ordinals we prove (in $\mathbf{ZFC}$) that the assertion "every $S$-ladder system has $\aleph_0$-uniformization" is equivalent to "every strongly $\aleph_1$-free abelian group of cardinality…

An abelian group is said to be aleph_1-free if all its countable subgroups are free. Our main result is: If R is a ring with R^+ free and |R|<lambda <= 2^{aleph_0}, then there exists an aleph_1-free abelian group G of cardinality lambda…

It is proved consistent with either CH or the negation of CH that there is an aleph_1-separable group of cardinality aleph_1 which does not have a coherent system of projections. It had previously been shown that it is consistent with not…

We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective…

In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which $\aleph_{\omega+1}$ carries a uniform ultrafilter that is $\theta$-indecomposable for every uncountable cardinal $\theta<\aleph_\omega$. In this…

We begin with the existence of groups with trivial duals for cardinals aleph_n (n in omega). Then we derive results about strongly aleph_n-free abelian groups of cardinality aleph_n (n in omega) with prescribed free, countable endomorphism…

It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a…

It is proved that, on any Abelian group of infinite cardinality ${\bf m}$, there exist precisely $2^{2^{\bf m}}$ nonequivalent bounded Hausdorff group topologies. Under the continuum hypothesis, the number of nonequivalent compact and…

An E-ring is a unital ring R such that every endomorphism of the underlying abelian group R^+ is multiplication by some ring-element. The existence of almost-free E-rings of cardinality greater than 2^{aleph_0} is undecidable in ZFC. While…

We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [EM],(p.453). There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly lambda-free ones…

Answering a problem of Eklof and Mekler, we show that there is a reflexive group of cardinality >= first measurable.

We show that if 2^{aleph_0} Cohen reals are added to the universe, then for every reduced non-free torsion-free abelian group A of cardinality less than the continuum, there is a prime p so that Ext_p(A, Z) not= 0. In particular if it is…

We investigate the Whiteheadness of Borel abelian groups (aleph_1-free, wlog, as otherwise this is trivial). We show that CH (and even WCH) implies any such abelian group is free, and always aleph_2-free.

Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals $\lambda$ such that there is a universal abelian $p$-group for purity of cardinality $\lambda$, i.e., an abelian $p$-group $U_\lambda$ of cardinality $\lambda$ such that every…

We study ultrafilters on regular uncountable cardinals, with a primary focus on $\omega_1$, and particularly in relation to the Tukey order on directed sets. Results include the independence from ZFC of the assertion that every uniform…

One of the better-known independence results in general mathematics is Shelah's solution to Whitehead's problem of whether $\mathrm{Ext}^1(A,\mathbb{Z})=0$ implies that an abelian group $A$ is free. The point of departure for the present…