General Non-structure Theory

TL;DR

This paper develops a general framework for constructing complex models from index models like linear orders and trees, using properties like bigness and reflection to analyze stability and diversity of models.

Contribution

It introduces a unified non-structure theory for building and analyzing models from various index structures, including new conditions for complexity and diversity.

Findings

Constructs models reflecting index structures such as linear orders and trees.

Provides conditions under which models are highly complex or rigid.

Shows that many non-isomorphic models exist for certain classes, especially linear orders.

Abstract

The theme of the first two sections, is to prepare the framework of how from a "complicated" family of index models I in K_1 we build many and/or complicated structures in a class K_2. The index models are characteristically linear orders, trees with kappa+1 levels (possibly with linear order on the set of successors of a member) and linearly ordered graph, for this we phrase relevant complicatedness properties (called bigness). We say when M in K_2 is represented in I in K_1. We give sufficient conditions when {M_I:I\in K^1_\lambda} is complicated where for each I in K^1_lambda we build M_I in K^2 (usually in K^2_lambda) represented in it and reflecting to some degree its structure (e.g. for I a linear order we can build a model of an unstable first order class reflecting the order). If we understand enough we can even build e.g. rigid members of K^2_lambda. Note that we mention "stable", "superstable", but in a self contained way, using an equivalent definition which is useful here and explicitly given. We also frame the use of generalizations of Ramsey and Erdos-Rado theorems to get models in which any I from the relevant K_1 is reflected. We give in some detail how this may apply to the class of separable reduced Abelian p-group and how we get relevant models for ordered graphs (via forcing). In the third section we show stronger results concerning linear orders. If for each linear order I of cardinality lambda>aleph_0 we can attach a model M_I in K_lambda in which the linear order can be embedded so that for enough cuts of I, their being omitted is reflected in M_I, then there are 2^lambda non-isomorphic cases.