Toward categoricity for classes with no maximal models

DOI10.1016/S0168-0072(98)00015-3zbMath0945.03048arXivmath/9707227MaRDI QIDQ1302297

Saharon Shelah, Andrés Villaveces

Publication date: 8 October 2000

Published in: Annals of Pure and Applied Logic (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/9707227

zbMATH Keywords

stabilityamalgamationsaturated modelscategoricity spectrumabstract elementary classesclassification theory

Mathematics Subject Classification ID

Classification theory, stability, and related concepts in model theory (03C45) Properties of classes of models (03C52) Categoricity and completeness of theories (03C35)

Related Items (38)

Shelah's eventual categoricity conjecture in universal classes. I. ⋮ Canonical forking in AECs ⋮ CATEGORICITY FROM ONE SUCCESSOR CARDINAL IN TAME ABSTRACT ELEMENTARY CLASSES ⋮ Building independence relations in abstract elementary classes ⋮ Beginning of stability theory for Polish spaces ⋮ Abstract elementary classes stable in \(\aleph_{0}\) ⋮ Superstability and symmetry ⋮ Toward a stability theory of tame abstract elementary classes ⋮ Good frames in the Hart-Shelah example ⋮ ON CATEGORICITY IN SUCCESSIVE CARDINALS ⋮ Saturation and solvability in abstract elementary classes with amalgamation ⋮ On universal modules with pure embeddings ⋮ On the uniqueness property of forking in abstract elementary classes ⋮ STABILITY RESULTS ASSUMING TAMENESS, MONSTER MODEL, AND CONTINUITY OF NONSPLITTING ⋮ Shelah's eventual categoricity conjecture in tame abstract elementary classes with primes ⋮ EQUIVALENT DEFINITIONS OF SUPERSTABILITY IN TAME ABSTRACT ELEMENTARY CLASSES ⋮ Superstability, Noetherian rings and pure-semisimple rings ⋮ Upward categoricity from a successor cardinal for tame abstract classes with amalgamation ⋮ Shelah's eventual categoricity conjecture in universal classes. II ⋮ Superstability from categoricity in abstract elementary classes ⋮ Chains of saturated models in AECs ⋮ Symmetry in abstract elementary classes with amalgamation ⋮ Forking in short and tame abstract elementary classes ⋮ Tameness and extending frames ⋮ TAMENESS FROM LARGE CARDINAL AXIOMS ⋮ Notes on Quasiminimality and Excellence ⋮ Categoricity in abstract elementary classes with no maximal models ⋮ Downward categoricity from a successor inside a good frame ⋮ Symmetry and the union of saturated models in superstable abstract elementary classes ⋮ FORKING AND SUPERSTABILITY IN TAME AECS ⋮ Shelah's categoricity conjecture from a successor for tame abstract elementary classes ⋮ Limit models in metric abstract elementary classes: the categorical case ⋮ Uniqueness of limit models in classes with amalgamation ⋮ On superstability in the class of flat modules and perfect rings ⋮ The categoricity spectrum of large abstract elementary classes ⋮ Algebraic description of limit models in classes of abelian groups ⋮ Categoricity for abstract classes with amalgamation ⋮ Toward categoricity for classes with no maximal models

Cites Work

• Classification theory for non-elementary classes. I: The number of uncountable models of \(\psi \in L_{\omega _ 1,\omega}\)
• Categoricity of theories in \(L_{\kappa \omega}\), with \(\kappa\) a compact cardinal
• Existence of many \(L_{\infty,\lambda}\)-equivalent, non-isomorphic models of T of power \(\lambda\)
• Classification theory and the number of non-isomorphic models
• Toward categoricity for classes with no maximal models
• Proper and Improper Forcing
• Categoricity of an abstract elementary class in two successive cardinals
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