# Whitehead groups may not be free even assuming ch, II

@article{Shelah1980WhiteheadGM,
title={Whitehead groups may not be free even assuming ch, II},
author={Saharon Shelah},
journal={Israel Journal of Mathematics},
year={1980},
volume={35},
pages={257-285}
}
• S. Shelah
• Published 1 December 1980
• Mathematics
• Israel Journal of Mathematics
AbstractWe prove some theorems on uncountable abelian groups, and consistency results promised in the first part, and also that a variant of $$\diamondsuit _{\omega _1 }$$ called ♣ (club), is consistent with 2ℵ0
92 Citations
On uncountable abelian groups
We continue the investigation from [10], [11], [12] on uncountable abelian groups. This paper tends more to group theory and was motivated by Nunke’s statement (in [9]) that Whitehead problem,Expand
A combinatorial principle equivalent to the existence of non-free Whitehead groups
• Mathematics
• 1994
As a consequence of identifying the principle described in the title, we prove that for any uncountable cardinal lambda, if there is a lambda-free Whitehead group of cardinality lambda which is notExpand
On the Existence of Rigid ℵ1-Free Abelian Groups of Cardinality ℵ1
• Mathematics
• 1995
An abelian group is said to be ℵ1-free if all its countable subgroups are free. A crucial special case of our main result can be stated immediately.
Every coseparable group may be free
• Mathematics
• 1993
AbstractWe show that if $$2^{\aleph _0 }$$ Cohen reals are added to the universe, then for every reduced non-free torsion-free abelian groupA of cardinality less than the continuum, there is aExpand
Uniformization and the diversity of Whitehead groups
• Mathematics
• 1992
Techniques of uniformization are used to prove that it is not consistent that the Whitehead groups of cardinality ℵ1 are exactly the strongly ℵ1-free groups. Some consequences of the assumption thatExpand
Filtration-equivalent aleph1-separable abelian groups of cardinality aleph1
• Mathematics, Computer Science
• Ann. Pure Appl. Log.
• 2010
Abstract We show that it is consistent with ordinary set theory Z F C and the generalized continuum hypothesis that there exist two ℵ 1 -separable abelian groups of cardinality ℵ 1 which areExpand
Filtration-equivalent א 1-separable abelian groups of cardinality א 1
We show that it is consistent with ordinary set theory ZFC and the generalized continuum hypothesis that there exist two א1-separable abelian groups of cardinality א1 which are filtration-equivalentExpand
Filtration equivalent aleph_1-separable abelian groups of cardinality aleph_1
• Mathematics
• 2006
We show that it is consistent with ordinary set theory ZFC and the generalized continuum hypothesis that there exist two separable abelian groups of cardinality aleph_1 which are filtrationExpand
The consistency of Ext(G, Z)=Q
For abelian groups, ifV=L, Ext(G, Z) cannot have cardinality ℵ0. We show that G.C.H. does not imply this. See Hiller and Shelah [2], Hiller, Huber and Shelah [3], Nunke [5] and Shelah [6, 7, 8] forExpand
Hedetniemi's conjecture for uncountable graphs
It is proved that in Godel's constructible universe, for every infinite successor cardinal k, there exist graphs G and H of size and chromatic number k, for which the tensor product graph (G x H) isExpand

#### References

SHOWING 1-10 OF 10 REFERENCES
Whitehead groups may be not free, even assuming CH, I
AbstractWe prove the consistency with ZFC+G.C.H. of an assertion, which implies several consequences of $$MA + 2^{\aleph _0 } > \aleph _1$$ , which $$\diamondsuit \aleph _1$$ implies theirExpand
Infinite abelian groups, whitehead problem and some constructions
We solve here some problems from Fuchs’ book. We show that the answer to Whitehead’s problem (for groups of power ℵ1) is independent from the usual axioms of set theory. We prove the existence ofExpand
On uncountable abelian groups
We continue the investigation from [10], [11], [12] on uncountable abelian groups. This paper tends more to group theory and was motivated by Nunke’s statement (in [9]) that Whitehead problem,Expand
Singular cohomology inL
• Mathematics
• 1977
We prove, that in the world of constructible sets, there does not exist a spaceX withH″ (X,Z) isomorphic to the rational numbers. The proof requires a result about the growth of ExtzEmphasis>/i(-, Z)Expand
A compactness theorem for singular cardinals, free algebras, Whitehead problem and tranversals
We prove, in an axiomatic way, a compactness theorem for singular cardinals. We apply it to prove that, for singular λ, every λ-free algebra is free; and similar compactness results for transversalsExpand
A new class of order types
Abstract Let φ 4 be the class of all order-types ϕ with the properties that every uncountable subtype of ϕ contains an uncountable well-ordering, but ϕ is not the union of countably manyExpand
A weak version of ◊ which follows from 2ℵ0<2ℵ1
• Mathematics
• 1978
We prove that if CH holds (or even if 2ℵ0 < 2ℵ1), then a weak version of ◊ holds. This weak version of ◊ is a ◊-like principle, and is strong enough to yield some of the known consequences of ◊.
Aspects of Constructibility
Zermelo-Fraenkel set theory.- The constructible universe.- The axiom of constructibility, the condensation lemma, and the consistency of the generalised continuum hypothesis.- The SouslinExpand