Whitehead groups may be not free, even assuming CH, I

  title={Whitehead groups may be not free, even assuming CH, I},
  author={Saharon Shelah},
  journal={Israel Journal of Mathematics},
  • S. Shelah
  • Published 1977
  • Mathematics
  • Israel Journal of Mathematics
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 their negation. 
Whitehead groups may not be free even assuming ch, II
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), isExpand
The consistency with CH of some consequences of Martin’s axiom plus $$2^{\aleph _0 } > \aleph _1 $$
We present here a (weak) axiom which implies some of the consequences of MA, but is consistent with GCH. We use the method of Jensen in his proof of consis (ZFC+GCH+SH).
Filtration-equivalent aleph1-separable abelian groups of cardinality aleph1
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
Uniformization and the diversity of Whitehead groups
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 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
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
The monadic theory of (ω2, <) may be complicated
SummaryAssume ZFC is consistent then for everyB⫅ω there is a generic extension of the ground world whereB is recursive in the monadic theory ofω2.
A model with Suslin trees but no minimal uncountable linear orders other than $\omega_1$ and $-\omega_1$
We show that the existence of a Suslin tree does not necessarily imply that there are uncountable minimal linear orders other than $\omega_1$ and $-\omega_1$, answering a question of J. Baumgartner.Expand
The uniformization property for ℵ2
AbstractWe present S. Shelah’s result thatS12={δ<ω2: cf(δ)=ω1} may have the uniformization property (cf., §1, or [3] for a definition) for “well-chosen sequences”, 〈ηδ:δ∈S12^ηδ anExpand
Iterated souslin forcing, the principles ⋄(E) and a generalisation of the axiom SAD
AbstractThe axiom SAD was introduced in our paper with Avraham and Shelah [1]. It is a Martin’s Axiom type of principle, having some of the consequences of MA plus $$2^{\aleph _0 } > \aleph _1 $$ ,Expand


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
The Souslin problem
Preliminaries.- Souslin's hypothesis.- The combinatorial property ?.- Homogeneous souslin trees and lines.- Rigid souslin trees and lines.- Martin's axiom and the consistency of SH.- TowardsExpand
A weak version of ◊ which follows from 2ℵ0<2ℵ1
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 ◊.
What is a Group Ring
(1976). What is a Group Ring? The American Mathematical Monthly: Vol. 83, No. 3, pp. 173-185.
Iterated Cohen extensions and Souslin's problem*
We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a countable denseExpand
Set Mappings, Partitions, and Chromatic Numbers
Publisher Summary This chapter discusses some aspects of combinatorial set theory related to recent results. The chapter presents results in their simplest forms so that their proofs reflect the mainExpand
Internal cohen extensions