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Categoricity and amalgamation for AEC and $ κ$ measurable

Oren Kolman, Saharon Shelah

Abstract

In the original version of this paper, we assume a theory $T$ that the logic $\mathbb L_{κ, \aleph_{0}}$ is categorical in a cardinal $λ> κ$, and $κ$ is a measurable cardinal. There we prove that the class of model of $T$ of cardinality $<λ$ (but $\geq |T|+κ$) has the amalgamation property; this is a step toward understanding the character of such classes of models. In this revised version we replaced the class of models of $T$ by $\mathfrak k$, an AEC (abstract elementary class) which has LS-number ${<} \, κ,$ or at least which behave nicely for ultrapowers by $D$, a normal ultra-filter on $κ$. Presently sub-section \S1A deals with $T \subseteq \mathbb L_{κ^{+}, \aleph_{0}}$ (and so does a large part of the introduction and little in the rest of \S1), but otherwise, all is done in the context of AEC.

Categoricity and amalgamation for AEC and $ κ$ measurable

Abstract

In the original version of this paper, we assume a theory that the logic is categorical in a cardinal , and is a measurable cardinal. There we prove that the class of model of of cardinality (but ) has the amalgamation property; this is a step toward understanding the character of such classes of models. In this revised version we replaced the class of models of by , an AEC (abstract elementary class) which has LS-number or at least which behave nicely for ultrapowers by , a normal ultra-filter on . Presently sub-section \S1A deals with (and so does a large part of the introduction and little in the rest of \S1), but otherwise, all is done in the context of AEC.

Paper Structure

This paper contains 10 sections, 30 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. Frame for $\mathbb{L}_{\kappa, \aleph_{0}}$
  4. Replacing $T$ by AEC
  5. Indiscernibles and Ehrenfeucht-Mostowski structures
  6. Limit ultra-powers, iterated ultra-powers and nice extensions
  7. The amalgamation property for regular categoricity
  8. Towards removing the assumption of regularity from the existence of universal extensions
  9. $(\theta, \partial)$-saturated models
  10. THE AMALGAMATION PROPERTY FOR $\mathfrak{k}_{< \lambda}$

Key Result

Theorem 1

Suppose that $T$ is a theory in a fragment of $\mathbb{L}_{\kappa, \aleph_{0}}$ where $\kappa$ is a measurable cardinal. If $T$ is categorical in the cardinal $\lambda > \kappa + | T |$, then ${\mathcal{K}}_{< \lambda}$, the class of models of $T$ of power strictly less than $\lambda$ (but $\geq \ch

Theorems & Definitions (117)

  • Theorem 1
  • Theorem 1.3
  • proof : Proof.
  • Definition 1.7
  • Definition 1.12
  • Example 1.13
  • Example 1.14
  • Definition 1.15
  • Definition 1.17
  • Definition 1.18
  • ...and 107 more