Guido Gherardi and Alberto Marcone

How Incomputable is the Separable Hahn-Banach Theorem?

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function Sep and a natural notion of reducibilit...

How constructive is constructing measures?

We investigate variations on the problem to construct a measure with a given support. The variations include the available information about the set, whether the support has to be precisely the given set or merely a subset, and whether the measure is required to be non-atomic. A special case of particular importance is the Frostman lemma, which links having certain Hausdorff dimension to admitt...

An Analysis of the Lemmas of Urysohn and Urysohn-Tietze According to Effective Borel Measurability

The Bolzano-Weierstrass Theorem is the Jump of Weak König's Lemma

We classify the computational content of the Bolzano-Weierstraß Theorem and variants thereof in the Weihrauch lattice. For this purpose we first introduce the concept of a derivative or jump in this lattice and we show that it has some properties similar to the Turing jump. Using this concept we prove that the derivative of closed choice of a computable metric space is the cluster point problem...

Reverse mathematics, well-quasi-orders, and Noetherian spaces