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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 of cardinals satisfying (where Π is the set of all primes), there is a torsion-free abelian group G of size ν such that equals the p-rank of Ext(G,ℤ) for every...
We show that log is needed to eliminate quantifiers in the theory of the real numbers with restricted analytic functions and exponentiation.
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
order preserving automorphisms of pairwise distinct growth rates.
We show that for no infinite group the class of abelian-by- groups is elementary, but, at least when is an infinite elementary abelian -group (with prime), the class of groups admitting a normal abelian subgroup whose quotient group is elementarily equivalent to is elementary.
In this paper we introduce a new invariant for extensions of difference fields, the distant degree, and discuss its properties.
(1) Shepherdson proved that a discrete unitary commutative semi-ring
A+ satisfies IE0 (induction scheme restricted to quantifier
free formulas) iff A is integral part of a
real closed field; and Berarducci asked about extensions of this
criterion when exponentiation is added to the language of rings. Let T range over axiom systems for ordered fields with
exponentiation; for three values of T we provide a theory
in the language of rings plus exponentiation such that the
...
A long-standing conjecture of Podewski states that every minimal field is algebraically closed. Known in positive characteristic, it remains wide open in characteristic zero. We reduce Podewski's conjecture to the (partially) ordered case, and we conjecture that such fields do not exist. We prove the conjecture in case the incomparability relation is transitive (the almost linear case).
We also study minimal groups with a (partial) order, and give a complete classification of...
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