Abstract
AbstractWe present finitary formulations of two well known results concerning infinite series, namely Abel's theorem, which establishes that if a series converges to some limit then its Abel sum converges to the same limit, and Tauber's theorem, which presents a simple condition under which the converse holds. Our approach is inspired by proof theory, and in particular Gödel's functional interpretation, which we use to establish quantitative versions of both of these results.
Highlights
In an essay of 2007 [14] T
The equivalence of Cauchy convergence and metastability is established via purely logical reasoning, and as was quickly observed, the correspondence principle as presented in [14] has deep connections with proof theory
We provide new quantitative versions of these two theorems, which take the shape of a route between various forms of metastability
Summary
In an essay of 2007 [14] (later published as part of [15]) T. The infinitary theorems can be directly derived from ours in a uniform way, using purely logical reasoning Though both Abel’s and Tauber’s theorems are elementary to state and prove, establishing in each case a natural finitary formulation from which the original theorem can be rederived is non-trivial, as is generally the case when it comes to correctly finitizing infinitary statements (an illuminating discussion of the subtleties which arise from the elementary infinite pigeonhole principle is given in [3]). The first is the fact that Abelian and Tauberian theorems give rise to simple and yet illuminating examples of the correspondence principle and related concepts such as metastability, which can be presented in such a way that we are not required to explicitly introduce any proof theoretic concepts (even the notion of a higher order functional is only needed in Section 5 to rederive the original results). We conjecture that a wealth of interesting case studies for applied proof theory can be found in this area, and hope that in this article to have taken a first step in this direction
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
