  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.
