On unsuperstable theories in GDST

TL;DR

This paper investigates the complexity of classifying models of different first-order theories, showing that unsuperstable theories have more complex isomorphism relations than classifiable ones under certain set-theoretic assumptions.

Contribution

It establishes a strict hierarchy in the $orel$-reducibility of isomorphism relations between classifiable and unsuperstable theories using coloured trees.

Findings

Unsupersable theories have more complex isomorphism relations than classifiable theories.

Under certain cardinality assumptions, the isomorphism relation of unsuperstable theories is strictly above that of classifiable theories.

The study uses coloured trees to analyze the $orel$-reducibility of these relations.

Abstract

We study the $\kappa$-Borel-reducibility of isomorphism relations of complete first order theories by using coloured trees. Under some cardinality assumptions, we show the following: For all theories T and T', if T is classifiable and T' is unsuperstable, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to $\kappa$-Borel-reducibility.