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- Ulrich Kohlenbach, Alexander P. Kreuzer
- Notre Dame Journal of Formal Logic
- 2009

This paper addresses the strength of Ramseyâ€™s theorem for pairs (RT2) over a weak base theory from the perspective of â€˜proof miningâ€™. Let RT2âˆ’ 2 denote Ramseyâ€™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 KÃ¶nigâ€™s lemma and a substantialâ€¦ (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 Î 2-conservative over ACA0 and thus that ACA0 +(U) is conservative over PA. Moreover, weâ€¦ (More)

- Vasco Brattka, Matthew Hendtlass, Alexander P. Kreuzer
- Theory of Computing Systems
- 2017

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,â€¦ (More)

- Ulrich Kohlenbach, Alexander P. Kreuzer
- J. Symb. Log.
- 2012

In this paper we study with proof-theoretic methods the function(al)s provably recursive relative to Ramseyâ€™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 + Î 1-CP are primitive recursive. This also provides a uniform method to extract bounds from proofs that useâ€¦ (More)

- Alexander P. Kreuzer
- Math. Log. Q.
- 2014

- Alexander P. Kreuzer
- J. Comb. Theory, Ser. A
- 1995

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 inâ€¦ (More)

- Alexander P. Kreuzer
- Logical Methods in Computer Science
- 2014

We analyze the strength of Hellyâ€™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 theâ€”so calledâ€”weakâˆ— topology on BV . We obtain that HST is instance-wiseâ€¦ (More)

- Alexander P. Kreuzer
- J. Logic & Analysis
- 2012

Let (X , d) be a complete metric space, m âˆˆ N \ {0}, and Î³ âˆˆ R with 0 â‰¤ Î³ < 1. A g-contraction is a mapping T : X âˆ’â†’ X such that for all x, y âˆˆ X there is an i âˆˆ [1,m] with d(T ix,T iy) <R Î³id(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 Ramseyâ€™sâ€¦ (More)

- Alexander P. Kreuzer
- J. Comb. Theory, Ser. A
- 1993

- Alexander P. Kreuzer
- J. Symb. Log.
- 2015

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 Hindmanâ€™s theorem (IHT) and we show that ACA 0 + (Uidem) is Î 2-conservativeâ€¦ (More)