In this paper we calibrate the exact proof-theoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction.

In ordinal analysis of impredicative theories so-called collapsing functions are of central importance. Unfortunately, the definition procedure of these functions makes essential use of uncountable cardinals whereas the notation system that they call into being corresponds to a recursive ordinal. It has long been claimed that, instead, one should manage to develop such functions directly on the basis of admissible ordinals. This paper is meant to show how this can be done. Interpreting the collapsing functions as operating directly on admissible (...) sets also renders a new and perspicuous approach to well-ordering proofs possible. MSC: 03F15, 03F35. (shrink)

Reinhardt 's strong mechanistic thesis, a formalization of “I know I am a Turing machine”, is shown to be consistent with Epistemic Arithmetic.

We will study patterns which occur when considering how Σ 1 -elementary substructures arise within hierarchies of structures. The order in which such patterns evolve will be seen to be independent of the hierarchy of structures provided the hierarchy satisfies some mild conditions. These patterns form the lowest level of what we call patterns of resemblance . They were originally used by the author to verify a conjecture of W. Reinhardt concerning epistemic theories 449–460; Ann. Pure Appl. Logic, to appear), (...) but their relationship to axioms of infinity and usefulness for ordinal analysis were manifest from the beginning. This paper is the first part of a series which provides an introduction to an extensive program including the ordinal analysis of set theories. Future papers will conclude the introduction and establish, among other things, that notations we will derive from the patterns considered here represent the proof-theoretic ordinal of the theory KPℓ 0 or, equivalently, Π 1 1 −CA 0. (shrink)

We will introduce a partial ordering $\preceq_1$ on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use $\preceq_1$ to provide a new characterization of the ubiquitous ordinal $\epsilon _{0}$.

In the first part we show why ordinals and ordinal notations are naturally connected with proof theoretical research. We introduce the program of ordinal analysis. The second part gives examples of applications of ordinal analysis.

We will study patterns which occur when considering how Σ1-elementary substructures arise within hierarchies of structures. The order in which such patterns evolve will be seen to be independent of the hierarchy of structures provided the hierarchy satisfies some mild conditions. These patterns form the lowest level of what we call patterns of resemblance. They were originally used by the author to verify a conjecture of W. Reinhardt concerning epistemic theories 449–460; Ann. Pure Appl. Logic, to appear), but their relationship (...) to axioms of infinity and usefulness for ordinal analysis were manifest from the beginning. This paper is the first part of a series which provides an introduction to an extensive program including the ordinal analysis of set theories. Future papers will conclude the introduction and establish, among other things, that notations we will derive from the patterns considered here represent the proof-theoretic ordinal of the theory KPℓ0 or, equivalently, Π11−CA0. (shrink)

Abstract.This paper is the second in a series of three culminating in an ordinal analysis of Π12-comprehension. Its objective is to present an ordinal analysis for the subsystem of second order arithmetic with Δ12-comprehension, bar induction and Π12-comprehension for formulae without set parameters. Couched in terms of Kripke-Platek set theory, KP, the latter system corresponds to KPi augmented by the assertion that there exists a stable ordinal, where KPi is KP with an additional axiom stating that every set is contained (...) in an admissible set. (shrink)

The exact correspondence between ordinal notations derived from Skolem hull operators, which are classical in ordinal analysis, and descriptions of ordinals in terms of Σ1-elementarity, an approach developed by T.J. Carlson, is analyzed in full detail. The ordinal arithmetical tools needed for this purpose were developed in [G. Wilken, Ordinal arithmetic based on Skolem hulling, Annals of Pure and Applied Logic 145 130–161]. We show that the least ordinal κ such that κ<1∞ 19–77] and described below) is the proof theoretic (...) ordinal of the set-theoretic system , confirming a claim of Carlson. Moreover, we characterize the class of all ordinals κ such that κ<1∞ and provide an ordinal arithmetical analysis of Carlson’s entire structure in the style of [T.J. Carlson, Ordinal arithmetic and Σ1-elementarity, Archive for Mathematical Logic 38 449–460]. (shrink)