A simple continuous theory

TL;DR

This paper explores continuous first-order theories, introducing a stronger condition to identify simple theories that do not interpret infinite discrete structures, with examples including richly branching $\

Contribution

It presents a new stronger condition for continuous theories and provides examples of simple and superstable theories without infinite discrete structures.

Findings

A stronger condition helps identify simple continuous theories without infinite discrete structures.

Richly branching $\

A superstable theory can lack this stronger condition yet still not interpret infinite discrete structures.

Abstract

In the context of continuous first-order logic, special attention is often given to theories that are somehow continuous in an 'essential' way. A common feature of such theories is that they do not interpret any infinite discrete structures. We investigate a stronger condition that is easier to establish and use it to give an example of a strictly simple continuous theory that does not interpret any infinite discrete structures: the theory of richly branching $\mathbb{R}$-forests with generic binary predicates. We also give an example of a superstable theory that fails to satisfy this stronger condition but nevertheless does not interpret any infinite discrete structures.