In this note the well-ordering principle for the derivative \ of normal functions \ on ordinals is shown to be equivalent to the existence of arbitrarily large countable coded \-models of the well-ordering principle for the function \.

The paper contains proof-theoretic investigation on extensions of Kripke-Platek set theory, KP, which accommodate first-order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Пn reflection rules. This leads to consistency proofs for the theories KP+Пn reflection using a small amount of arithmetic and the well-foundedness of a certain ordinal system with respect to primitive decending sequences. Regarding future work, we intend to avail ourselves of these new cut (...) elimination techniques to attain an ordinal analysis of П12 comprehension by approaching П12 comprehension through transfinite levels of reflection. (shrink)

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle and I (...) apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions. (shrink)

A recursive notation system of a strong segment of ordinals was developped by Jäger [3]. An unessential modified versionT(J) of this notation system was described in [4]. In the following, the well-ordering ofT(J) is proved in a formal system of second order arithmetic with the axiom schema ofΠ 2 1 -comprehension. It follows, that the proof theoretical ordinal ofΠ 2 1 -analysis is greater than the order type ofT(J).