Ein in der reinen Zahlentheorie unbeweisbarer Satz über endliche Folgen von natürlichen Zahlen

A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets

“…In concrete situations W may for example stand for an iterated application of basic constructions like disjoint union and cartesian product, the set of finite sequences construction, the multiset construction, or a tree constructor and the like. We assume that for W we have an explicit knowledge of o(W (X)) such that o(W (X)) = o(W (o(X))) and such that this equality can be proved using an effective reification (An example for this technique is, for example, given in [13] or [16]). …”

“…This step uses the assumption that the maximal order type resulting from W can be computed by an effective reification a la [13] or [16]. Therefore the definition of ϑ yields…”

Worms, gaps, and hydras

“…We show that reduction sequences of worms correspond to bad sequences of finite sequences of natural numbers with respect to Schütte-Simpson's notion of gap-embeddability (see [11]), which is the one-dimensional version of Friedman's notion of gap-embedding of finite trees. We infer some Friedman-style independence results.…”

“…We infer some Friedman-style independence results. We also present a characterization of Schütte-Simpson's strong gap-embeddability (see [11]) in terms of provability logic due to Lev Beklemishev.…”

An Inductive Version of Nash-Williams’ Minimal-Bad-Sequence Argument for Higman’s Lemma

“…A variant of this proof was translated by Murthy via Friedman's A-translation into a constructive proof [Mur91], however resulting in a huge proof whose computational content couldn't yet be discovered. More direct constructive proofs were given by Schütte/Simpson [SS85], Murthy/Russell [MR90], and Richman/Stolzenberg [RS93]. The Schütte/Simpson proof uses ordinal notations up to 0 and is related to an earlier proof by Schmidt [Sch79], the other proofs are carried out in a (proof theoretically stronger) theory of inductive definitions.…”