Results 1  10
of
15
Analytic and pseudoanalytic structures
 Proc. Logic Colloquium
, 2000
"... One of the questions frequently asked nowadays about model theory is whether it is still logic. The reason for asking the question is mainly that more and more of model theoretic research focuses on concrete mathematical fields, uses extensively their tools and attacks their inner problems. Neverthe ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
(Show Context)
One of the questions frequently asked nowadays about model theory is whether it is still logic. The reason for asking the question is mainly that more and more of model theoretic research focuses on concrete mathematical fields, uses extensively their tools and attacks their inner problems. Nevertheless the logical roots in the case of model theoretic geometric stability theory are not only clear but also remain very important in all its applications. This line of research started with the notion of a κcategorical first order theory, which quite soon mutated into the more algebraic and less logical notion of a κcategorical structure. A structure M in a first order language L is said to be categorical in cardinality κ if there is exactly one, up to isomorphism, structure of cardinality κ satisfying the Ltheory of M. In other words, if we add to Th(M) the (non firstorder) statement that the cardinality of the domain of the structure is κ, the description becomes categorical. The principal breakthrough, in the midsixties, from which stability theory started
What is a Structure Theory
 Bulletin of the London Mathematical Society
, 1987
"... into the 'Mathematical logic and foundations ' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consis ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
into the 'Mathematical logic and foundations ' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consisted of equalities and inequalities between uncountable cardinals. In my experience most mathematicians find uncountable cardinals depressing, if they have any reaction to them at all. In fact Shelah was quite right to be so happy, but not because of his cardinal inequalities. He had just brought to a successful conclusion a line of research which had cost him fourteen years of intensive work and not far off a hundred published books and papers. In the course of this work he had established a new range of questions about mathematics with implications far beyond mathematical logic. That is what I want to discuss here. In brief, Shelah's work is about the notion of a class of structures which has a good structure theory. We all have a rough intuitive notion of what counts as a good structure theory. For example the structure theory of finitely generated abelian groups
DECIDABILITY OF THE THEORY OF MODULES OVER COMMUTATIVE VALUATION DOMAINS
"... Abstract. We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all Vmodules is decidable. 1. ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all Vmodules is decidable. 1.
First order theory of cyclically ordered groups, submitted, hall00879429, version 1 or arXiv:1311.0499v1
"... ar ..."
(Show Context)
Relative categoricity in abelian groups II
, 2008
"... We consider structures A consisting of an abelian group with a subgroup AP distinguished by a 1ary relation symbol P, and complete theories T of such structures. Such a theory T is (κ, λ)categorical if T has models A of cardinality λ with AP  = κ, and given any two such models A, B with AP = BP ..."
Abstract
 Add to MetaCart
We consider structures A consisting of an abelian group with a subgroup AP distinguished by a 1ary relation symbol P, and complete theories T of such structures. Such a theory T is (κ, λ)categorical if T has models A of cardinality λ with AP  = κ, and given any two such models A, B with AP = BP, there is an isomorphism from A to B which is the identity on AP. We classify all complete theories of such structures A in terms of the cardinal pairs (κ, λ) in which they are categorical. We classify algebraically the A of finite order λ with AP of order κ which are (κ, λ)categorical.
Conclusions References
, 2002
"... 3D air pollution modelling L. M. Frohn et al. Validation of a 3D hemispheric nested air ..."
Abstract
 Add to MetaCart
3D air pollution modelling L. M. Frohn et al. Validation of a 3D hemispheric nested air