Alexander P. Kreuzer

NON-PRINCIPAL ULTRAFILTERS, PROGRAM EXTRACTION AND HIGHER-ORDER REVERSE MATHEMATICS

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher-order arithmetic. Let be the statement that a non-principal ultrafilter on ℕ exists and let be the higher-order extension of ACA0. We show that is -conservative over and thus that is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly statement ∀ f ∃ g Aqf(f, g) in a realizing term in Godels system T can be extracted. This means that one can extract a term t ∈ T, such that ∀ f Aqf(f, t(f)).

On the Uniform Computational Content of Computability Theory

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.

On principles between ∑1- and ∑2-induction, and monotone enumerations

We show that many principles of first-order arithmetic, previously only known to lie strictly between ∑1-induction and ∑2-induction, are equivalent to the well-foundedness of ωω. Among these principles are the iteration of partial functions (P∑1) of Hajek and Paris, the bounded monotone enumerations principle (non-iterated, BME1) by Chong, Slaman, and Yang, the relativized Paris–Harrington principle for pairs, and the totality of the relativized Ackermann–Peter function. With this we show that the well-foundedness of ωω is a far more widespread than usually suspected. Further, we investigate the k-iterated version of the bounded monotone iterations principle (BMEk), and show that it is equivalent to the well-foundedness of the (k + 1)-height ω-tower ω⋰ω.

ON IDEMPOTENT ULTRAFILTERS IN HIGHER-ORDER REVERSE MATHEMATICS

We analyze the strength of the existence of idempotent ultrafilters in higher-order reverse mathematics. Let

From Bolzano‐Weierstraß to Arzelà‐Ascoli

\left( {{{\cal U}_{{\rm{idem}}}}} \right)

Bounded variation and the strength of Helly's selection theorem

be the statement that an idempotent ultrafilter on ℕ exists. We show that over

From Bolzano-Weierstra{\ss} to Arzel\`a-Ascoli

ACA_0^\omega

Measure theory and higher order arithmetic

, the higher-order extension of ACA 0 , the statement

A lower bound on Gowers' FIN_k theorem

\left( {{{\cal U}_{{\rm{idem}}}}} \right)

On principles between

implies the iterated Hindman’s theorem (IHT) and we show that