Proof-theoretic analysis of KPM

@article{Rathjen1991ProoftheoreticAO,
title={Proof-theoretic analysis of KPM},
author={Michael Rathjen},
journal={Archive for Mathematical Logic},
year={1991},
volume={30},
pages={377-403}
}
• M. Rathjen
• Published 1 September 1991
• Mathematics
• Archive for Mathematical Logic
AbstractKPM is a subsystem of set theory designed to formalize a recursively Mahlo universe of sets. In this paper we show that a certain ordinal notation system is sufficient to measure the proof-theoretic strength ofKPM. This involves a detour through an infinitary calculus RS(M), for which we prove several cutelimination theorems. Full cut-elimination is available for derivations of $$\Sigma (L_{\omega _1^c } )$$ sentences, whereω1c denotes the least nonrecursive ordinal. This paper is self…
85 Citations
Proof Theory of Reflection
Inaccessibility in Constructive Set Theory and Type Theory
• Mathematics, Philosophy
Ann. Pure Appl. Log.
• 1998
Ordinal diagrams for recursively Mahlo universes
• T. Arai
• Computer Science, Mathematics
Arch. Math. Log.
• 2000
It is shown that for each $\alpha<\Omega$ in $O(\mu)$KPM proves that the initial segment of $N$ determined by $\ alpha$ is a well ordering.
Proof theory and Martin-Löf Type Theory
It is the goal of ordinal theoretic proof theory to reduce the consistency of theories for formalising mathematical proofs to the well-foundedness of ordinal notation systems. In order to obtain a
Wellordering proofs for metapredicative Mahlo
• T. Strahm
• Mathematics
Journal of Symbolic Logic
• 2002
It is shown that in the corresponding theories EMA of explicit mathematics and KPm0 of admissible set theory, transfinite induction along initial segments of the ordinal φω00, for φ being a ternary Veblen function, is derivable.
The strength of some Martin-Löf type theories
• Mathematics
Arch. Math. Log.
• 1994
The determination of the proof-theoretic strength of Martin-Löf's type theory with a universe and the type of well-founded trees is determined, showing that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic.
Proof theory of Martin-Löf type theory: An overview
We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert’s program, ordinal theoretic
An ordinal analysis of stability
An ordinal representation system based on ν-indescribable cardinals is introduced which is then employed for determining an upper bound for the proof–theoretic strength of the theory KPi+ ∀ρ ∃ππ is π+ρ-stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.
Well-Ordering Principles in Proof Theory and Reverse Mathematics
Several theorems about the equivalence of familiar theories of reverse mathematics with certain well-ordering principles have been proved by recursion-theoretic and combinatorial methods (Friedman,

References

SHOWING 1-10 OF 17 REFERENCES
Proof Theory: An Introduction
This book contains the somewhat extended lecture notes of an introductory course in proof theory. As for the choice about the parts to he presented the anthor opts for what he considers to be the
ϱ-inaccessible ordinals, collapsing functions and a recursive notation system
The uniform definition procedure of this paper is possible to define the functions O ° for arbitrary O and will not work with higher O-functions but with the functions T,.
Proof theory and ordinal analysis
In the first part, it is shown why ordinals and ordinal notations are naturally connected with proof theoretical research and the program of ordinal analysis is introduced.
Nichtbeweisbarkeit von gewissen kombinatorischen Eigenschaften endlicher Bäume
These results provide the most dramatic examples so far known of mathematically meaningful theorems of finite combinatorics which are unprovable in certain logical systems.