# Metastable convergence and logical compactness

## TL;DR

This paper characterizes the logical frameworks where the Uniform Metastability Principle applies, showing it holds precisely in compact logics, and provides new insights into logical compactness and metastable convergence.

## Contribution

It precisely identifies the logical frameworks for which the UMP holds, linking it to the concept of logical compactness and offering new characterizations.

## Key findings

- UMP holds iff the logic is compact
- Topological version of the equivalence established
- New characterizations of logical compactness provided

## Abstract

The concept of metastable convergence was identified by Tao;it allows converting theorems about convergence into stronger theorems about uniform convergence. The Uniform Metastability Principle (UMP) states that if $T$ is a theorem about convergence, then the fact that $T$ is valid implies automatically that its (stronger) uniform version is valid, provided that $T$ can be stated in certain logical frameworks. In this paper we identify precisely the logical frameworks $L$ for which UMP holds. More precisely, we prove that the UMP holds for $L$ if and only if $L$ is a compact logic. We also prove a topological version of this equivalence. We conclude by proving new characterizations of logical compactness that yield additional information about the UMP.