Gunnar Wilken

The Bachmann-Howard Structure in Terms of Σ1-Elementarity

The Bachmann-Howard structure, that is the segment of ordinal numbers below the proof theoretic ordinal of Kripke-Platek set theory with infinity, is fully characterized in terms of CARLSON’s approach to ordinal notation systems based on the notion of Σ1-elementarity.

Tracking chains of Σ2-elementarity☆

We apply the ordinal arithmetical tools that were developed in Wilken (2007) [10] and Carlson and Wilken (in press) [4] in order to introduce tracking chains as the crucial means in the arithmetical analysis of (pure) elementary patterns of resemblance of order 2; see Carlson (2001) [2], Carlson (2009) [3], and Carlson and Wilken (in preparation) [5]. Although underlying heuristics for an analysis of Σ2-elementarity within the structure R2 is given in [5], this article is independent of [5] and provides a complete arithmetical analysis of the structure R2 below the least ordinal α such that any pure pattern of order 2 has a covering below α. α is shown to be the proof-theoretic ordinal of KPl0.

Ordinal arithmetic with simultaneously defined theta-functions

This article provides a detailed comparison between two systems of collapsing functions. These functions play a crucial role in proof theory, in the analysis of patterns of resemblance, and the analysis of maximal order types of well partial orders. The exact correspondence given here serves as a starting point for far reaching extensions of current results on patterns and well partial orders

Pure patterns of order 2

Abstract We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order 2, showing that the latter characterize the proof-theoretic ordinal 1 ∞ of the fragment Π 1 1 – CA 0 of second order number theory, or equivalently the set theory KP l 0 . As a corollary, we prove that Carlsons result on the well-quasi orderedness of respecting forests of order 2 implies transfinite induction up to the ordinal 1 ∞ . We expect that our approach will facilitate analysis of more powerful systems of patterns.

Derivation lengths classification of Gödel's T extending Howard's assignment

Let T be Godels system of primitive recursive functionals of finite type in the lambda formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of natural numbers such that if a term a reduces to a term b and if a natural number I(a) is assigned to a then a natural number I(b) can be assigned to b such that I(a) is greater than I(b). The construction of I is based on Howards 1970 ordinal assignment for T and Weiermanns 1996 treatment of T in the combinatory logic version. As a corollary we obtain an optimal derivation length classification for the lambda formulation of T and its fragments. Compared with Weiermanns 1996 exposition this article yields solutions to several non-trivial problems arising from dealing with lambda terms instead of combinatory logic terms. It is expected that the methods developed here can be applied to other higher order rewrite systems resulting in new powerful termination orderings since T is a paradigm for such systems.

Goodstein sequences for prominent ordinals up to the ordinal of Π11−CA0

Abstract We introduce strong Goodstein principles which are true but unprovable in strong impredicative theories like ID n .

Complexity of Godel's T in lambda-formulation

Let T be Godels system of primitive recursive functionals of finite type in the lambda -formulation. We define by constructive means using recursion on nested multisets a multivalued function / from the set of terms of T into the set of natural numbers such that if a term a reduces to a term b and if a natural number 1(a) is assigned to a then a natural number 1(b) can be assigned to 6 such that 1(a) > 1(b). The construction of / is based on Howards 1970 ordinal assignment for T and Weiermanns 1996 treatment of T in the combinatory logic version. As a corollary we obtain an optimal derivation length classification for the lambda -formulation of T and its fragments. Compared with Weiermanns 1996 exposition this article yields solutions to several non-trivial problems arising from dealing with lambda -terms instead of combinatory logic terms.

Complexity of Gödel's T in λ-Formulation

Let T be Godels system of primitive recursive functionals of finite type in the lambda -formulation. We define by constructive means using recursion on nested multisets a multivalued function / from the set of terms of T into the set of natural numbers such that if a term a reduces to a term b and if a natural number 1(a) is assigned to a then a natural number 1(b) can be assigned to 6 such that 1(a) > 1(b). The construction of / is based on Howards 1970 ordinal assignment for T and Weiermanns 1996 treatment of T in the combinatory logic version. As a corollary we obtain an optimal derivation length classification for the lambda -formulation of T and its fragments. Compared with Weiermanns 1996 exposition this article yields solutions to several non-trivial problems arising from dealing with lambda -terms instead of combinatory logic terms.