Publicaties
Chapter 7 : Intermediate Goodstein principles Universiteit Gent
The original Goodstein process proceeds by writing natural numbers in nested exponential k-normal form, then successively raising the base to k + 1 and subtracting one from the end result. Such sequences always reach zero, but this fact is unprovable in Peano arithmetic. In this chapter, we instead consider notations for natural numbers based on the Ackermann function. We define three new Goodstein processes, obtaining new independence results ...
Arithmetical and hyperarithmetical worm battles Universiteit Gent
Japaridze’s provability logic (GLP) has one modality [n] for each natural number and has been used by Beklemishev for a proof theoretic analysis of Peano arithmetic (PA) and related theories. Among other benefits, this analysis yields the so-called Every Worm Dies (EWD) principle, a natural combinatorial statement independent of PA. Recently, Beklemishev and Pakhomov have studied notions of provability corresponding to transfinite modalities in ...
Goodstein sequences on a parametrized Ackermann-Peter function Universiteit Gent
Following our [6], though with somewhat different methods here, further variants of Goodstein sequences are introduced in terms of parameterized AckermannU+2013Péter functions. Each of the sequences is shown to terminate, and the proof-theoretic strengths of these facts are calibrated by means of ordinal assignments, yielding independence results for a range of theories: PRA, PA, U+03A311 -DC 0 , ATR 0 , up to ID 1 . The key is the so-called ...
Calculating maximal order types for finite rooted unstructured labeled trees Universiteit Gent
Diana Schmidt, in her Habilitationsschrift in 1979, completely classified the maximal order types of the natural tree embeddability relations for finite rooted structured labeled trees. Her results since have found interesting applications in proof theory and reverse mathematics. The question concerning the maximal order types of unstructured trees has been left open for years, and a conclusive answer will be given in this article. Moreover, we ...