Full Amalgamation Classes with Intrinsic Transcendentals

TL;DR

This paper investigates full amalgamation classes with intrinsic transcendentals, showing they often have undecidable theories but can also be strictly superstable or simple, with complexity arising from intrinsic types.

Contribution

It introduces new results on the model-theoretic properties of classes with intrinsic transcendentals, including undecidability and stability classifications, and analyzes the complexity of types.

Findings

Generic models can have finite subsets with intrinsic closure not in algebraic closure.

Under natural conditions, the theories are essentially undecidable.

Examples of strictly superstable and strictly simple classes are provided.

Abstract

We develop some basic results about full amalgamation classes with intrinsic trascendentals. These classes have generics whose models may have finite subsets whose intrinsic closure is not contained in its algebraic closure. We will show that under fairly natural conditions the generic will have an essentially undecidable theory, but we will also exhibit strictly superstable and strictly simple examples. Separating types over a model into those that are intrinsic and those that are extrinsic, we will demonstrate that the complexity exceeding that of a simple theory in the classes with essentially undecidable theories of [1] comes from the intrinsic types by deriving a class from them which has a strictly simple theory with few intrinsic types. 1. J. Brody and M.C. Laskowski, "On Rational Limits of Shelah-Spencer Graphs", Journal of Symbolic Logic 77 (2012), no. 2, 580-592