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 U be the statement that a non-principal ultrafilter exists and let ACA_0^{\omega} be the higher order extension of ACA_0. We show that ACA_0^{\omega}+U is \Pi^1_2-conservative over ACA_0^{\omega} and thus that ACA_0^{\omega}+\U is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly \Pi^1_2 statement \forall f \exists g A(f,g) in ACA_0^{\omega}+U a realizing term in G\"odel's system T can be extracted. This means that one can extract a term t, such that A(f,t(f))

Weakly o-minimal structures and Skolem functions

The monotonicity theorem is the first step in proving that o-minimal structures satisfy cellular decomposition, which gives a comprehensive picture of the definable subsets in an o-minimal structure. This leads to the fact that any o-minimal structure has an o-minimal theory. We first investigate the possible analogues for monotonicity in a weakly o-minimal structure, and find that having definable Skolem functions and uniform elimination of imaginaries is sufficient to guarantee that a weakly o-minimal theory satisfies one of these, the Finitary Monotonicity Property. In much of the work on weakly o-minimal structures, it is shown that nonvaluational weakly o-minimal structures are most "like" the o-minimal case. To that end, there is a monotonicity theorem and a strong cellular decomposition for nonvaluational weakly o-minimal expansions of a group. In contrast to these results, we show that nonvaluational weakly o-minimal expansions of an o-minimal group do not have definable Skolem functions. As a partial converse, we show that certain valuational expansions of an o-minimal group, called T-immune, do have definable Skolem functions, and we calculate them explicitly via quantifier elimination

The cohesive principle and the Bolzano-Weierstra{\ss} principle

The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstra{\ss} principle (BW) and a weak variant of it. We show that BW is instance-wise equivalent to the weak K\"onig's lemma for $\Sigma^0_1$-trees ($\Sigma^0_1$-WKL). This means that from every bounded sequence of reals one can compute an infinite $\Sigma^0_1$-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.Comment: corrected typos, slightly improved presentatio

Panorama of p-adic model theory

ABSTRACT. We survey the literature in the model theory of p-adic numbers since\ud Denef’s work on the rationality of Poincaré series. / RÉSUMÉ. Nous donnons un aperçu des développements de la théorie des modèles\ud des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poincaré,\ud par une revue de la bibliographie

A constructive interpretation of Ramsey's theorem via the product of selection functions

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