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Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are… (More)

- P. Aczel, M. Rathjen
- 2008

4 Operations on Sets and Classes 25 4.1 Class Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Class Relations and Functions . . . . . . . . . . . . . . . . . . 26 4.3 Some Consequences of Union-Replacement . . . . . . . . . . . 27 4.4 Russell’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.5 Subset Collection and… (More)

- Michael Rathjen
- Arch. Math. Log.
- 1991

- Edward R. Griffor, Michael Rathjen
- Arch. Math. Log.
- 1994

One objective of this paper is the determination of the proof{theoretic strength of Martin{ LL of's type theory with a universe and the type of well{founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with 1 2 comprehension and bar induction. As… (More)

- Michael Rathjen, Andreas Weiermann
- Ann. Pure Appl. Logic
- 1993

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.

- Michael Rathjen
- 1997

A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their ‘rank’ or ‘complexity’ in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of ‘proof theoretic ordinals’ to… (More)

- MICHAEL RATHJEN
- 1995

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- Michael Rathjen
- 2004

The intent of this paper is to study generalized inductive definitions on the basis of Constructive Zermelo-Fraenkel Set Theory, CZF. In theories such as classical Zermelo-Fraenkel Set Theory, it can be shown that every inductive definition over a set gives rise to a least and a greatest fixed point, which are sets. The latter principle, notated GID, can… (More)

- Michael Rathjen
- Ann. Pure Appl. Logic
- 1994

The paper contains proof{theoretic investigations on extensions of Kripke{Platek set theory, KP, which accommodate rst order re ection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for in nitary calculi of rami ed set theory with n re ection rules. This leads to consistency proofs for the theories KP+ n{re ection… (More)

- M. RATHJEN
- 2003

The paper investigates the strength of the anti-foundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in diiering elds such as mathematical logic, computer science , artiicial intelligence, linguistics, cognitive science, and philosophy.… (More)