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31
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
"... 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 sup ..."
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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.
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 fo ..."
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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.
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
Absolutely Rigid Systems and Absolutely Indecomposable Groups
, 2004
"... We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecomposable, that is, they remain indecomposable in any generic extension. However, any absolutely rigid family of groups has cardinality less than the partition car ..."
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We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecomposable, that is, they remain indecomposable in any generic extension. However, any absolutely rigid family of groups has cardinality less than the partition cardinal κ(ω). Added December 2004: The proofs of Theorems 0.2 and 0.3 are not correct, and the claimed results remain open. (The ”only if ” part assertion in the last 3 lines before ”Proof of (II) ” on p. 266 is not correct.) However, Theorems 0.1 and 0.4, which give upper bounds to the size of rigid systems/groups, are valid. And the construction in the proof of Theorem 0.3 does yield an affirmative answer to Nadel’s question whether there is a proper class of torsionfree abelian groups which are pairwise absolutely nonisomorphic. 0