We 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.

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The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstraß principle and a weak variant of it.We show that BW is instance-wise equivalent to the weak König’s lemma for Σ01-trees . This means that from every bounded sequence of reals one can compute an infinite Σ01-0/1-tree, such that each infinite branch of it yields an accumulation point and vice versa. Especially, this shows that the degrees d ≫ 0′ are exactly those containing (...) an accumulation point for all bounded computable sequences.Let BWweak be the principle stating that every bounded sequence of real numbers contains a Cauchy subsequence . We show that BWweak is instance-wise equivalent to the cohesive principle and—using this—obtain a classification of the computational and logical strength of BWweak. Especially we show that BWweak does not solve the halting problem and does not lead to more than primitive recursive growth. Therefore it is strictly weaker than BW. We also discuss possible uses of BWweak. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

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In this paper we study with proof-theoretic methods the function(al) s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functional provably recursive from $RCA_0 + COH + \Pi _1^0 - CP$ are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that $WKL_0 + \Pi _1^0 - (...) CP + COH$ is $\Pi _1^0 $ -conservative over PRA. Recent work of the first author showed that $\Pi _1^0 + CP + COH$ is equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey's theorem for pairs and two colors $\left( {RT_2^2 } \right)$ we obtain the upper bounded that the type 2 functionals provable recursive relative to $RCA_0 + \sum\nolimits_2^0 {IA + RT_2^2 } $ are in T₁. This is the fragment of Gödel's system T containing only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T₁-bounds from proofs that use RT2..... Moreover, this yields a new proof of the fact that $WKL_0 + \sum\nolimits_2^0 {IA} + RT_2^2 $ is $\Pi _1^0 $ -conservative over $RCA_0 + \sum\nolimits_2^0 { - IA} $ . The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using $\Pi _1^0 $ -comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for $RT_2^2 $. (shrink)

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We prove a result relating the author's monotone functional interpretation to the bounded functional interpretation due to Ferreira and Oliva. More precisely we show that largely a solution for the bounded interpretation also is a solution for the monotone functional interpretation although the latter uses the existence of an underlying precise witness. This makes it possible to focus on the extraction of bounds while using the conceptual benefit of having precise realizers at the same time without having to construct them.

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In recent years, proof theoretic transformations that are based on extensions of monotone forms of Gödel’s famous functional interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content consists both in effective quantitative bounds as well as in qualitative uniformity results. One of the main ineffective tools in abstract functional analysis is the use of sequential forms of weak compactness. As we recently verified, the sequential form of weak compactness for bounded closed and (...) convex subsets of an abstract Hilbert space can be carried out in suitable formal systems that are covered by existing metatheorems developed in the course of the proof mining program. In particular, it follows that the monotone functional interpretation of this weak compactness principle can be realized by a functional Ω∗ definable from bar recursion of lowest type. While a case study on the analysis of strong convergence results that are based on weak compactness indicates that the use of the latter seems to be eliminable, things apparently are different for weak convergence theorems such as the famous Baillon nonlinear ergodic theorem. For this theorem we recently extracted an explicit bound on a metastable version of this theorem that is primitive recursive relative to Ω∗.In the current paper we for the first time give the construction of Ω∗ . As a corollary to the fine analysis of the use of bar recursion in this construction we obtain that Ω∗ elevates arguments in Tn at most to resulting functionals in Tn+2 . In particular, one can conclude from this that our bound on Baillon’s theorem is at least definable in T4. (shrink)

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