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A characterization of Ext(G,ℤ) assuming (V = L)

Saharon Shelah, Lutz Strüngmann (2007)

Fundamenta Mathematicae

We complete the characterization of Ext(G,ℤ) for any torsion-free abelian group G assuming Gödel’s axiom of constructibility plus there is no weakly compact cardinal. In particular, we prove in (V = L) that, for a singular cardinal ν of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence ( ν p : p Π ) of cardinals satisfying ν p 2 ν (where Π is the set of all primes), there is a torsion-free abelian group G of size ν such that ν p equals the p-rank of Ext(G,ℤ) for every...

A failure of quantifier elimination.

Angus Macintyre, David Marker (1997)

Revista Matemática de la Universidad Complutense de Madrid

We show that log is needed to eliminate quantifiers in the theory of the real numbers with restricted analytic functions and exponentiation.

A family of 2 1 logarithmic functions of distinct growth rates

Salma Kuhlmann (2010)

Open Mathematics

We construct a totally ordered set Γ of positive infinite germs (i.e. germs of positive real-valued functions that tend to +∞), with order type being the lexicographic product ℵ1 × ℤ2. We show that Γ admits 2 1 order preserving automorphisms of pairwise distinct growth rates.

An elementary class extending abelian-by- G groups, for G infinite

Carlo Toffalori (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We show that for no infinite group G the class of abelian-by- G groups is elementary, but, at least when G is an infinite elementary abelian p -group (with p prime), the class of groups admitting a normal abelian subgroup whose quotient group is elementarily equivalent to G is elementary.

An invariant for difference field extensions

Zoé Chatzidakis, Ehud Hrushovski (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper we introduce a new invariant for extensions of difference fields, the distant degree, and discuss its properties.

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