# Definable Continuous Induction on Ordered Abelian Groups

## TL;DR

This paper develops a first-order definable version of continuous induction on densely ordered abelian groups, enabling a proof of a definable Heine-Borel theorem within this algebraic structure.

## Contribution

It introduces a novel, first-order definable form of continuous induction for ordered abelian groups, extending the classical real analysis tools to algebraic structures.

## Key findings

- Proves Heine-Borel theorem in densely ordered abelian groups.
- Introduces pseudo finite sets for definability.
- Establishes a definable version of continuous induction.

## Abstract

As mathematical induction is applied to prove statements on natural numbers, {\it continuous induction} (or, {\it real induction}) is a tool to prove some statements in real analysis.(Although, this comparison is somehow an overstatement.) Here, we first consider it on densely ordered abelian groups to prove Heine-Borel theorem (every closed and bounded interval is compact with respect to order topology) in those structures. Then, using the recently introduced notion of pseudo finite sets, we introduce a first order definable version of {\it continuous induction} in the language of ordered groups and we use it to prove a definable version of Heine-Borel theorem on densely ordered abelian groups.