Upward categoricity from a successor cardinal for tame abstract classes with amalgamation

2005 ◽  
Vol 70 (2) ◽  
pp. 639-660 ◽  
Author(s):  
Olivier Lessmann
Keyword(s):  
Elementary Class ◽  
Uncountable Cardinal ◽  
Abstract Elementary Class

AbstractThis paper is devoted to the proof of the following upward categoricity theorem: Let be a tame abstract elementary class with amalgamation, arbitrarily large models, and countable Löwenheim-Skolem number. If is categorical in ℵ then is categorical in every uncountable cardinal. More generally, wc prove that if is categorical in a successor cardinal λ+ then is categorical everywhere above λ+.

AN ABSTRACT ELEMENTARY CLASS NONAXIOMATIZABLE IN

2019 ◽  
Vol 84 (3) ◽  
pp. 1240-1251
Author(s):  
SIMON HENRY
Keyword(s):  
Regular Cardinal ◽  
Vector Spaces ◽  
Left Adjoint ◽  
Scott Topology ◽  
Elementary Class ◽  
Closed Fields ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Bounded Below ◽  
Abstract Elementary Class

AbstractWe show that for any uncountable cardinal λ, the category of sets of cardinality at least λ and monomorphisms between them cannot appear as the category of points of a topos, in particular is not the category of models of a ${L_{\infty ,\omega }}$-theory. More generally we show that for any regular cardinal $\kappa < \lambda$ it is neither the category of κ-points of a κ-topos, in particular, nor the category of models of a ${L_{\infty ,\kappa }}$-theory.The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least λ and monomorphisms between them. The same techniques also apply to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree.

scholarly journals N⊥ as an abstract elementary class

2007 ◽  
Vol 149 (1-3) ◽  
pp. 25-39 ◽  
Author(s):  
John T. Baldwin ◽  
Paul C. Eklof ◽  
Jan Trlifaj
Keyword(s):  
Elementary Class ◽  
Abstract Elementary Class

Independence, dimension and continuity in non-forking frames

2013 ◽  
Vol 78 (2) ◽  
pp. 602-632 ◽  
Author(s):  
Adi Jarden ◽  
Alon Sitton
Keyword(s):  
Continuity Property ◽  
Natural Conditions ◽  
Elementary Class ◽  
Independence Dimension ◽  
The Stability ◽  
Abstract Elementary Class

AbstractThe notion J is independent in (M, M0, N) was established by Shelah, for an AEC (abstract elementary class) which is stable in some cardinal λ and has a non-forking relation, satisfying the good λ-frame axioms and some additional hypotheses. Shelah uses independence to define dimension.Here, we show the connection between the continuity property and dimension: if a non-forking satisfies natural conditions and the continuity property, then the dimension is well-behaved.As a corollary, we weaken the stability hypothesis and two additional hypotheses, that appear in Shelah's theorem.

scholarly journals Shelah's categoricity conjecture from a successor for tame abstract elementary classes

2006 ◽  
Vol 71 (2) ◽  
pp. 553-568 ◽  
Author(s):  
Rami Grossberg ◽  
Monica Vandieren
Keyword(s):  
Abstract Elementary Classes ◽  
Transfer Theorem ◽  
Elementary Class ◽  
Joint Embedding ◽  
Elementary Classes ◽  
Abstract Elementary Class

AbstractWe prove a categoricity transfer theorem for tame abstract elementary classes.Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K+}. If K is categorical in λ and λ+, then K is categorical in λ++.Combining this theorem with some results from [37]. we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes:Suppose K is χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ0 ≔ Hanf(K). Ifand K is categorical in somethen K is categorical in μ for all μ .

scholarly journals FORKING AND SUPERSTABILITY IN TAME AECS

2016 ◽  
Vol 81 (1) ◽  
pp. 357-383 ◽  
Author(s):  
SEBASTIEN VASEY
Keyword(s):  
General Construction ◽  
Transfer Theorem ◽  
Elementary Class ◽  
Abstract Elementary Class ◽  
Limit Models

AbstractWe prove that any tame abstract elementary class categorical in a suitable cardinal has an eventually global good frame: a forking-like notion defined on all types of single elements. This gives the first known general construction of a good frame in ZFC. We show that we already obtain a well-behaved independence relation assuming only a superstability-like hypothesis instead of categoricity. These methods are applied to obtain an upward stability transfer theorem from categoricity and tameness, as well as new conditions for uniqueness of limit models.

scholarly journals ALMOST GALOIS ω-STABLE CLASSES

2015 ◽  
Vol 80 (3) ◽  
pp. 763-784 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
PAUL B. LARSON ◽  
SAHARON SHELAH
Keyword(s):  
Elementary Class ◽  
Joint Embedding ◽  
Image Position ◽  
Abstract Elementary Class ◽  
Countable Models

AbstractTheorem. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presentable abstract elementary class with Löwenheim–Skolem number ℵ0, satisfying the joint embedding and amalgamation properties in ℵ0. If K has only countably many models in ℵ1, then all are small. If, in addition, k is almost Galois ω-stable then k is Galois ω-stable. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presented almost Galois ω-stable AEC satisfying amalgamation for countable models, and having a model of cardinality ℵ1. The assertion that K is ℵ1-categorical is then absolute.

scholarly journals Notes on Quasiminimality and Excellence

2004 ◽  
Vol 10 (3) ◽  
pp. 334-366 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Special Kind ◽  
Order Logic ◽  
First Order Logic ◽  
Structure Theory ◽  
Mathematical Structures ◽  
First Order ◽  
Elementary Class ◽  
Abstract Elementary Class

AbstractThis paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for Lω1,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.

DISJOINT AMALGAMATION IN LOCALLY FINITE AEC

2017 ◽  
Vol 82 (1) ◽  
pp. 98-119 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
MARTIN KOERWIEN ◽  
MICHAEL C. LASKOWSKI
Keyword(s):  
Locally Finite ◽  
Elementary Class ◽  
Image Position ◽  
Abstract Elementary Class

AbstractWe introduce the concept of a locally finite abstract elementary class and develop the theory of disjoint$\left( { \le \lambda ,k} \right)$-amalgamation) for such classes. From this we find a family of complete ${L_{{\omega _1},\omega }}$ sentences ${\phi _r}$ that a) homogeneously characterizes ${\aleph _r}$ (improving results of Hjorth [11] and Laskowski–Shelah [13] and answering a question of [21]), while b) the ${\phi _r}$ provide the first examples of a class of models of a complete sentence in ${L_{{\omega _1},\omega }}$ where the spectrum of cardinals in which amalgamation holds is other that none or all.

scholarly journals MEASURING DEPENDENCE IN METRIC ABSTRACT ELEMENTARY CLASSES WITH PERTURBATIONS

2017 ◽  
Vol 82 (4) ◽  
pp. 1199-1228
Author(s):  
ÅSA HIRVONEN ◽  
TAPANI HYTTINEN
Keyword(s):  
Homogeneous Model ◽  
Usual Sense ◽  
Abstract Elementary Classes ◽  
Weyl Algebras ◽  
Elementary Class ◽  
Free Extension ◽  
Type Spaces ◽  
Elementary Classes ◽  
Measure Of Dependence ◽  
Abstract Elementary Class

AbstractWe define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is dp-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces and we give a new example of such a class based on B. Zilber’s approximations of Weyl algebras. We introduce a way to measure the dependence of a tuple a from a set B over another set A. We prove basic properties of the notion, e.g., that a is independent of B over A in the usual sense of homogeneous model theory if and only if the measure of dependence is < ε for all ε > 0. In well behaved situations, the measure corresponds to the distance to a free extension. As an example of our measure of dependence we show a connection between the measure and entropy in models from quantum mechanics in which the spectrum of the observable is discrete. As an application, we show that weak simplicity implies a very strong form of simplicity and study the question of when the dependence inside a set of all realisations of some type can be seen to arise from a pregeometry in cases when the type is not regular. In the end of the paper, we demonstrate our notions and results in one more example: a class built from the p-adic integers.

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