Metastable convergence and logical compactness
Xavier Caicedo, Eduardo Duéñez, José Iovino · 2023
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 https://www.w3.org/1998/Math/MathML" display="inline"> T https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_1.tif"/> is a theorem about convergence, then the fact that https://www.w3.org/1998/Math/MathML" display="inline"> T https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_2.tif"/> is valid implies automatically that its (stronger) uniform version is valid, provided that https://www.w3.org/1998/Math/MathML" display="inline"> T https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_3.tif"/> can be stated in certain logical frameworks. In this paper we identify precisely the logical frameworks for which UMP holds. More precisely, we prove that the UMP holds for in a logic https://www.w3.org/1998/Math/MathML" display="inline"> L https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_4.tif"/> if and only if https://www.w3.org/1998/Math/MathML" display="inline"> L https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_5.tif"/> 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.