We demonstrate that the Weihrauch lattice can be used to classify the uniform computational content of computability-theoretic properties as well as the computational content of theorems in one common setting. The properties that we study include diagonal non-computability, hyperimmunity, complete consistent extensions of Peano arithmetic, 1-genericity, Martin-Löf randomness, and cohesiveness. The theorems that we include in our case study are the low basis theorem of Jockusch and Soare, the Kleene-Post theorem, and Friedberg's jump inversion theorem. It turns out that all the aforementioned properties and many theorems in computability theory, including all theorems that claim the existence of some Turing degree, have very little uniform computational content: they are located outside of the upper cone of binary choice (also known as LLPO); we call problems with this property indiscriminative. Since practically all theorems from classical analysis whose computational content has been classified are discriminative, our observation could yield an explanation for why theorems and results in computability theory typically have very few direct consequences in other disciplines such as analysis. A notable exception in our case study is the low basis theorem which is discriminative. This is perhaps why it is considered to be one of the most applicable theorems in computability theory. In some cases a bridge between the indiscriminative world and the discriminative world of classical mathematics can be established via a suitable residual operation and we demonstrate this in the case of the cohesiveness problem and the problem of consistent complete extensions of Peano arithmetic. Both turn out to be the quotient of two discriminative problems.

Abstract. We analyze the strength of Helly's selection theorem (HST), which is the most important compactness theorem on the space of functions of bounded variation (BV ). For this we utilize a new representation of this space intermediate between L1 and the Sobolev space W 1,1 , compatible with the-so called-weak * topology on BV . We obtain that HST is instance-wise equivalent to the Bolzano-Weierstraß principle over RCA0. With this HST is equivalent to ACA0 over RCA0. A similar classification is obtained in the Weihrauch lattice.In this paper we investigate the space of functions of bounded variation (BV ) and Helly's selection theorem (HST) from the viewpoint of reverse mathematics and computable analysis. Helly's selection theorem is the most important compactness principle on BV . It is used in analysis and optimization, see for instance [1,3].This continues our work in [10] and [12] where (instances of) the Bolzano-Weierstraß principle and the Arzelà-Ascoli theorem were analyzed. There we showed, among others, that an instance of the Arzelà-Ascoli theorem is equivalent to a suitable single instance of the Bolzano-Weierstraß principle (for the unit interval [0, 1]), which, in turn, is equivalent to an instance of WKL for Σ 0 1 -trees. Here, we will show that an instance of Helly's selection theorem is equivalent to a single instance of the Bolzano-Weierstraß principle (and with this to an instance of the other principles mentioned above). It is a priori not clear that this is possible since the proof of HST uses seemingly iterated application of the Arzelà-Ascoli theorem and since there are compactness principles, which are instance-wise strictly stronger than Bolzano-Weierstraß for [0, 1]. (For instance the Bolzano-Weierstraß principle for weak compactness on ℓ 2 has this property, see [11].) A fortori this shows that HST is equivalent to ACA 0 over RCA 0 .

Abstract. We investigate the statement that the Lebesgue measure defined on all subsets of the Cantor-space exists. As base system we take ACA ω 0 + (µ). The system ACA ω 0 is the higher order extension of Friedman's system ACA 0 , and (µ) denotes Feferman's µ, that is a uniform functional for arithmetical comprehension defined by f (µ(f )) = 0 if ∃n f (n) = 0 for f ∈ N N . Feferman's µ will provide countable unions and intersections of sets of reals and is, in fact, equivalent to this. For this reason ACA ω 0 + (µ) is the weakest fragment of higher order arithmetic where σ-additive measures are directly definable.We obtain that over ACA ω 0 + (µ) the existence of the Lebesgue measure is Π 1 2 -conservative over ACA ω 0 and with this conservative over PA. Moreover, we establish a corresponding program extraction result.In this paper, we will investigate the statement that the Lebesgue measure, defined on all subsets of the Cantor-space 2 N , or equivalently the unit interval [0, 1], exists. The setting of this investigation will be higher order arithmetic-as base system we will take ACA ω 0 +(µ). This is the higher order extension of Friedman's system for arithmetical comprehension ACA 0 together with Feferman's µ, a functional which provides a uniform variant of arithmetical comprehension. The functional µ will be used to define countable unions and intersections of sets and is, in fact, equivalent to this. With this, ACA ω 0 + (µ) is the weakest system in which the textbook definition of measures, including σ-additivity, can be formulated. In addition to that it is strong enough to develop most of analysis since it contains ACA 0 , see [Sim09]. Moreover, in the same setting also other investigations on higher order reverse mathematics have been carried out, see for instance [Hun08] and [Sch13]. We therefore believe that ACA ω 0 + (µ) is a suitable system for investigating measure theory and the Lebesgue measure.Our main result is that ACA ω 0 + (µ) plus the existence of the Lebesgue measure defined on all subsets of the Cantor-space 2 N (denoted by (λ)) is Π

This paper addresses the strength of Ramsey's theorem for pairs (RT 2 2 ) over a weak base theory from the perspective of 'proof mining'. Let RT 2− 2 denote Ramsey's theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König's Lemma and a substantial amount of 0 1 -induction (enough to prove the totality of all primitive recursive functions but not of all primitive recursive functionals). In the resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of ∀∃-theorems.There are two components of this work. The first component is a general proof-theoretic result, due to the second author, that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey's theorem due to Erdős and Rado can be formalized using these restricted principles.So from the perspective of proof unwinding the computational content of concrete proofs based on RT 2 2 the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of RT 2 2 that are primitive recursive in f .

Abstract. The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstraß principle (BW) and a weak variant of it.We show that BW is instance-wise equivalent to the weak König's lemma for Σ 0 1 -trees (Σ 0 1 -WKL). This means that from every bounded sequence of reals one can compute an infinite Σ 0 1 -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 BW weak be the principle stating that every bounded sequence of real numbers contains a Cauchy subsequence (a sequence converging but not necessarily fast). We show that BW weak is instance-wise equivalent to the (strong) cohesive principle (StCOH) and -using this -obtain a classification of the computational and logical strength of BW weak . Especially we show that BW weak 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 BW weak .In this paper we investigate the logical and recursion theoretic strength of instances of the Bolzano-Weierstraß principle (BW) and the weak variant of it stating only the existence of a slow converging Cauchy subsequence (BW weak ). Slow converging means here that the rate of convergence does not need to be computable.Let weak König's lemma (WKL) be the principle stating that an infinite 0/1-tree has an infinite branch and let Σ 0 1 -WKL be the statement that an infinite 0/1-tree given by a Σ 0 1 -predicate has an infinite branch. We show that BW and Σ 0 1 -WKL are instance-wise equivalent. Instance-wise means here that for every instance of BW, i.e. every bounded sequence, one can compute, uniformly, an instance of Σ 0 1 -WKL, i.e. a code for an infinite Σ 0 1 -0/1-tree, such that from a solution of this instance of Σ 0 1 -WKL one can compute, uniformly, an accumulation point and vice versa. Instance-wise equivalence refines the usual logical equivalence where the full second order closure of the principles may be used -e.g. arithmetical comprehension (ACA 0 , i.e. the schema ∃X ∀n (n ∈ X ↔ φ(n)) for any arithmetical formula φ) and Π 0 1 -CA (comprehension where φ is restricted to Π 0 1 -formulas) are equivalent but they are not instance-wise equivalent. As consequence we obtain that the Turing degrees containing solutions to all instances of Σ 0 1 -WKL (i.e. the degrees d with d ≫ 0 ′ , see below) are exactly those containing an accumulation point for each computable bounded sequence.Furthermore, we show that BW weak is instance-wise equivalent to the strong cohesive principle, see Definition 1 below. Using this one can apply classification 2010 Mathematics Subject Classification. 03F60, 03D80, 03B30.