Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Limit Models in Strictly Stable Abstract Elementary Classes

Will Boney, Monica M. VanDieren

Abstract

In this paper, we examine the locality condition for non-splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove (note that no tameness is assumed): Suppose that $\mathcal{K}$ is an abstract elementary class satisfying 1. the joint embedding and amalgamation properties with no maximal model of cardinality $μ$. 2. stability in $μ$. 3. $κ^*_μ(\mathcal{K})<μ^+$. 4. continuity for non-$μ$-splitting (i.e. if $p\in\text{ga-S}(M)$ and $M$ is a limit model witnessed by $\langle M_i\mid i<α\rangle$ for some limit ordinal $α<μ^+$ and there exists $N \prec M_0$ so that $p\restriction M_i$ does not $μ$-split over $N$ for all $i<α$, then $p$ does not $μ$-split over $N$). For $θ$ and $δ$ limit ordinals $<μ^+$ both with cofinality $\geqκ^*_μ(\mathcal{K})$, if $\mathcal{K}$ satisfies symmetry for non-$μ$-splitting (or just $(μ,δ)$-symmetry), then, for any $M_1$ and $M_2$ that are $(μ,θ)$ and $(μ,δ)$-limit models over $M_0$, respectively, we have that $M_1$ and $M_2$ are isomorphic over $M_0$.

Limit Models in Strictly Stable Abstract Elementary Classes

Abstract

In this paper, we examine the locality condition for non-splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove (note that no tameness is assumed): Suppose that is an abstract elementary class satisfying 1. the joint embedding and amalgamation properties with no maximal model of cardinality . 2. stability in . 3. . 4. continuity for non--splitting (i.e. if and is a limit model witnessed by for some limit ordinal and there exists so that does not -split over for all , then does not -split over ). For and limit ordinals both with cofinality , if satisfies symmetry for non--splitting (or just -symmetry), then, for any and that are and -limit models over , respectively, we have that and are isomorphic over .

Paper Structure

This paper contains 5 sections, 6 theorems, 19 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Relatively Full Towers
  4. Reduced Towers
  5. Uniqueness of Long Limit Models

Key Result

Theorem 1.2

Suppose that $\mathcal{K}$ is an abstract elementary class satisfying For $\theta$ and $\delta$ limit ordinals $<\mu^+$ both with cofinality $\geq\kappa^*_\mu(\mathcal{K})$, if $\mathcal{K}$ satisfies symmetry for non-$\mu$-splitting (or just $(\mu,\delta)$-symmetry), then, for any $M_1$ and $M_2$ that are $(\mu,\theta)$ and $(\mu,\delta)$-limit models over $M_0$, res

Figures (3)

  • Figure 1: A diagram of the models and elements in the definition of $(\mu,\delta)$-symmetry. We assume the type $\mathop{\mathrm{ga-tp}}\nolimits(b/M)$ does not $\mu$-split over $M_0$ and $\mathop{\mathrm{ga-tp}}\nolimits(a/M_0)$ does not $\mu$-split over $N$. Symmetry implies the existence of $M^b$ a limit model over $M_0$ so that $\mathop{\mathrm{ga-tp}}\nolimits(a/M^b)$ does not $\mu$-split over $N$.
  • Figure 4: $(\bar{M},\bar{a},\bar{N})$ and the towers $(\bar{M},\bar{a},\bar{N})^i$ extending $(\bar{M},\bar{a},\bar{N})\restriction\delta$ that don't contain $b$.
  • Figure 5: The chain of length $\delta$ of towers of increasing index sets $I_j$ of cofinality $\theta+1$. The symbol $\lll$ indicates that there are $\mu$ many new indices between $i_\beta$ and $i_{\beta+1}$ in $I_{j+1}\backslash I_j$. The elements indexed by these indices realize all the strong types over the model $M^j_{i_\alpha}$. The notation $\prec_{u}$ is an abbreviation for a universal extension.

Theorems & Definitions (23)

  • Conjecture 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.4: Va1
  • Definition 2.5
  • proof
  • Definition 2.8
  • Remark 2.9
  • ...and 13 more