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Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
EVERY COTORSIONFREE RING IS AN Endomorphism Ring
, 1980
"... Some years ago A. L. S. Corner proved that every countable and cotorsionfree ring can be realized as the endomorphism ring of some torsionfree abelian group. This result has many interesting consequences for abelian groups. Using a settheoretic axiom Vk., which follows for instance from V = L, we ..."
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Some years ago A. L. S. Corner proved that every countable and cotorsionfree ring can be realized as the endomorphism ring of some torsionfree abelian group. This result has many interesting consequences for abelian groups. Using a settheoretic axiom Vk., which follows for instance from V = L, we can drop the countability condition in Corner's theorem.
On the existence of precovers
 BAER CENTENIAL VOLUME OF ILLINOIS J. MATH
"... It is proved consistent with ZFC + GCH that for every Whitehead group A of infinite rank, there is a Whitehead group HA such that Ext(HA, A) ̸ = 0. This is a strong generalization of the consistency of the existence of nonfree Whitehead groups. A consequence is that it is undecidable in ZFC + GCH ..."
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It is proved consistent with ZFC + GCH that for every Whitehead group A of infinite rank, there is a Whitehead group HA such that Ext(HA, A) ̸ = 0. This is a strong generalization of the consistency of the existence of nonfree Whitehead groups. A consequence is that it is undecidable in ZFC + GCH whether every Zmodule has a ⊥ {Z}precover. Moreover, for a large class of Zmodules N, it is proved consistent that a known sufficient condition for the existence of ⊥ {N}precovers is not satisfied.
Almost free splitters
 Colloquium Math
, 1999
"... Let R be a subring of the rationals. We want to investigate self splitting Rmodules G that is Ext R(G,G) = 0 holds. For simplicity we will call such modules splitters, see [16]. Also other names like stones are used, see a dictionary in Ringel’s paper [14]. Our investigation continues [10]. In [10 ..."
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Let R be a subring of the rationals. We want to investigate self splitting Rmodules G that is Ext R(G,G) = 0 holds. For simplicity we will call such modules splitters, see [16]. Also other names like stones are used, see a dictionary in Ringel’s paper [14]. Our investigation continues [10]. In [10] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsionfree, algebraically compact ones. In [10] we concentrated on splitters which are larger then the continuum and such that countable submodules are not necessarily free. The ‘opposite ’ case of ℵ1free splitters of cardinality less or equal to ℵ1 was singled out because of basically different techniques. This is the target of the present paper. If the splitter is countable, then it must be free over some subring of the rationals by Hausen [12]. Contrary to the results in [10] and in accordance to [12] we can show that all ℵ1free splitters of cardinality ℵ1 are free indeed. 1
The twocardinals transfer property and resurrection of supercompactness
 Proceedings of the American Mathematical Society
, 1996
"... Abstract. We show that the transfer property (ℵ1, ℵ0) → (λ +,λ) for singular λ does not imply (even) the existence of a nonreflecting stationary subset of λ +. The result assumes the consistency of ZFC with the existence of infinitely many supercompact cardinals. We employ a technique of “resurrec ..."
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Abstract. We show that the transfer property (ℵ1, ℵ0) → (λ +,λ) for singular λ does not imply (even) the existence of a nonreflecting stationary subset of λ +. The result assumes the consistency of ZFC with the existence of infinitely many supercompact cardinals. We employ a technique of “resurrection of supercompactness”. Our forcing extension destroys the supercompactness of some cardinals; to show that in the extended model they still carry some of their compactness properties (such as reflection of stationary sets), we show that their supercompactness can be resurrected via a tame forcing extension. 1.
On cardinalities in quotients of inverse limits of groups, Math Japonica 47
, 1998
"... Abstract. Let λ be ℵ0 or a strong limit of cofinality ℵ0. Suppose that 〈Gm, πm,n: m ≤ n < ω 〉 and 〈Hm, π t m,n: m ≤ n < ω 〉 are projective systems of groups of cardinality less than λ and suppose that for every n < ω there is a homorphism σ: Hn → Gn such that all the diagrams commute. If for every µ ..."
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Abstract. Let λ be ℵ0 or a strong limit of cofinality ℵ0. Suppose that 〈Gm, πm,n: m ≤ n < ω 〉 and 〈Hm, π t m,n: m ≤ n < ω 〉 are projective systems of groups of cardinality less than λ and suppose that for every n < ω there is a homorphism σ: Hn → Gn such that all the diagrams commute. If for every µ < λ there exists 〈fi ∈ Gω i ̸ = j = ⇒ fif −1 such that i ̸ = j = ⇒ fif −1 j: i < µ 〉 such that j ̸ ∈ σω(Hω) then there exists 〈fi ∈ Gω: i < 2 λ 〉 ̸ ∈ σω(Hω). 1.
Set Theory Generated by Abelian Group Theory
 Bull. Symbolic Logic
, 1997
"... Introduction. This survey is intended to introduce to logicians some notions, methods and theorems in set theory which aroselargely through the work of Saharon Shelahout of (successful) attempts to solve problems in abelian group theory, principally the Whitehead problem and the closely relate ..."
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Introduction. This survey is intended to introduce to logicians some notions, methods and theorems in set theory which aroselargely through the work of Saharon Shelahout of (successful) attempts to solve problems in abelian group theory, principally the Whitehead problem and the closely related problem of the existence of almost free abelian groups. While Shelah's first independence result regarding the Whitehead problem used established settheoretical methods (discussed below), his later work required new ideas; it is on these that we focus. We emphasize the nature of the new ideas and the historical context in which they arose, and we do not attempt to give precise technical definitions in all cases, nor to include a comprehensive survey of the algebraic results. In fact, very little algebraic background is needed beyond the definitions of group and group homomorphism. Unless otherwise specified, we will use the word "group" to refer to an abelian group, that is, the g
Topological characterization of torus groups
 Topology Appl
"... To the memory of Professor Ball Abstract. Topological characterization of torus groups is given. ..."
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To the memory of Professor Ball Abstract. Topological characterization of torus groups is given.
Examples of nonlocality
"... Elementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galoistypes. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on localit ..."
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Elementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galoistypes. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on locality: there is a transformation of AEC’s which preserves nonlocality but takes any AEC which admits intersections to one with amalgamation. More specifically we have: Theorem 5.3. There is an AEC with amalgamation which is not (ℵ0, ℵ1)tame but is (2 ℵ0, ∞)tame; Theorem 3.3. It is consistent with ZFC that there is an AEC with amalgamation which is not ( ≤ ℵ2, ≤ ℵ2)compact. A primary object of study in first order model theory is a syntactic type: the type of a over B in a model N is the collection of formulas φ(x, b) which are true of a in N. Usually the N is suppressed because a preliminary construction