Alexander P. Kreuzer

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This paper addresses the strength of Ramsey’s theorem for pairs (RT2) over a weak base theory from the perspective of ‘proof mining’. Let RT2− 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(More)
We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher order arithmetic. Let (U) be the statement that a non-principal ultrafilter on N exists and let ACA0 be the higher order extension of ACA0. We show that ACA0 + (U) is Π2-conservative over ACA0 and thus that ACA0 +(U) is conservative over PA. Moreover, we(More)
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,(More)
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 functionals provably recursive from RCA0 + COH + Π1-CP are primitive recursive. This also provides a uniform method to extract bounds from proofs that use(More)
A [0, m]-space is a linear space with the following property: For any point-line pair (x, G) there are at most m lines through x which are coplanar with G and which have no point in common with G. For every [0, m]-space (M, 90l) we define an order o r d M in a natural way. For dimM~>3 and o r d M ~ > 3 m + 2 , every [0, m]-space (M, 991) can be embedded in(More)
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 , compatible with the—so called—weak∗ topology on BV . We obtain that HST is instance-wise(More)
Let (X , d) be a complete metric space, m ∈ N \ {0}, and γ ∈ R with 0 ≤ γ < 1. A g-contraction is a mapping T : X −→ X such that for all x, y ∈ X there is an i ∈ [1,m] with d(T ix,T iy) <R γid(x, y). The generalized Banach contractions principle states that each g-contraction has a fixed point. We show that this principle is a consequence of Ramsey’s(More)
We analyze the strength of the existence of idempotent ultrafilters in higher-order reverse mathematics. Let (Uidem) be the statement that an idempotent ultrafilter on N exists. We show that over ACA 0 , the higher-order extension of ACA0, the statement (Uidem) implies the iterated Hindman’s theorem (IHT) and we show that ACA 0 + (Uidem) is Π2-conservative(More)