# Universal theories and compactly expandable models

## TL;DR

This paper explores the distinction between expandability and compact expandability in models, focusing on the logic of countable cofinality to deepen theoretical understanding.

## Contribution

It advances the theoretical framework distinguishing expandability types and investigates the logic of countable cofinality in model theory.

## Key findings

- Clarifies the difference between expandability and compact expandability.
- Provides new insights into the logic of countable cofinality.
- Enhances the theoretical understanding of model expandability.

## Abstract

Our aim is to solve a quite old question on the difference between expandability and compact expandability. Toward this, we further investigate the logic of countable cofinality.