
TL;DR
This paper investigates a hierarchy of convergence notions for sequences, revealing their stratification by countable ordinals and establishing strictness of these hierarchies through model theoretic and proof theoretic perspectives.
Contribution
It introduces two stratified notions of metastable convergence, shows their equivalence to existing concepts, and demonstrates the strictness of the hierarchy with explicit examples.
Findings
Uniform metastable convergence is equivalent to some alpha-uniform convergence.
Abstract omega-uniform convergence corresponds to bounded oscillation.
The hierarchy of convergence notions is strict, with explicit examples for each level.
Abstract
Uniform metastable convergence is a weak form of uniform convergence for a family of sequences. In this paper we explore the way that metastable convergence stratifies into a family of notions indexed by countable ordinals. We give two versions of this stratified family, loosely speaking, they correspond to the model theoretic and proof theoretic perspectives. For the model theoretic version, which we call abstract alpha-uniform convergence, we show that uniform metastable convergence is equivalent to abstract -uniform convergence for some alpha, and that abstract omega-uniform convergence is equivalent to uniformly bounded oscillation of the family of sequences. The proof theoretic version, which we call concrete alpha-uniform convergence, is less canonical (it depends on a choice of ordinal notation), but appears naturally when "proof mining" convergence proofs to obtain…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · Rings, Modules, and Algebras
