Results 1  10
of
11
Proof theory of reflection
 Annals of Pure and Applied Logic
, 1994
"... The paper contains proof–theoretic investigations 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 Πn reflection rules. Th ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
The paper contains proof–theoretic investigations 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 Πn reflection rules. This leads to consistency proofs for the theories KP + Πn–reflection using a small amount of arithmetic (PRA) and the well–foundedness of a certain ordinal notation system with respect to primitive recursive descending sequences. Regarding future work, we intend to avail ourselves of these new cut elimination techniques to attain an ordinal analysis of Π 1 2 comprehension by approaching Π1 2 comprehension through transfinite levels of reflection. 1
The Realm of Ordinal Analysis
 SETS AND PROOFS. PROCEEDINGS OF THE LOGIC COLLOQUIUM '97
, 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 ma ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
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 theories, gauging their `consistency strength' and `computational power'. Ordinaltheoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie "  the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency  technical results in pro...
An Ordinal Representation System for ...Comprehension and Related Systems
, 1995
"... The objective of this paper is to introduce an ordinal representation system which has been employed in the determination of the prooftheoretic strength of \Pi 1 2 comprehension and related systems. 1 Introduction The purpose of this paper is to provide an ordinal representation system for the s ..."
Abstract
 Add to MetaCart
The objective of this paper is to introduce an ordinal representation system which has been employed in the determination of the prooftheoretic strength of \Pi 1 2 comprehension and related systems. 1 Introduction The purpose of this paper is to provide an ordinal representation system for the system of \Pi 1 2 analysis, which is the subsystem of formal second order arithmetic, Z 2 , with comprehension confined to \Pi 1 2 formulae. The ordinal representation can also be used to provide ordinal analyses for theories that are reducible to iterated \Pi 1 2 comprehension, e.g. \Delta 1 3 comprehension. The details will be laid out in the second part of this paper. Ordinaltheoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z 2 . Considerable progress has been made...
Recent Advances In Ordinal Analysis: ... And Related Systems
, 1995
"... this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of # ..."
Abstract
 Add to MetaCart
this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of #
The proof theory of classical and constructive inductive definitions. A 40 year saga, 19682008.
, 2008
"... ..."
An Ordinal Analysis of parameter free ...Comprehension: Part I
"... The objective of this paper is to present an ordinal analysis for the fragment of second order arithmetic with \Delta 1 2 comprehension, bar induction and \Pi 1 2 comprehension for formulae without set parameters. ..."
Abstract
 Add to MetaCart
The objective of this paper is to present an ordinal analysis for the fragment of second order arithmetic with \Delta 1 2 comprehension, bar induction and \Pi 1 2 comprehension for formulae without set parameters.
An OrdinalFree Proof of the Cutelimination Theorem for an Impredicative Subsystem of
, 2009
"... The aim of this paper is to give a simple ordinalfree proof of the cutelimination theorem for an impredicative subsystem of Π 1 1analysis with ωrule containing ID1. A derivation d in this system is transformed into a derivation g(d) with Ωrule introduced by W. Buchholz in such a way that one cut ..."
Abstract
 Add to MetaCart
The aim of this paper is to give a simple ordinalfree proof of the cutelimination theorem for an impredicative subsystem of Π 1 1analysis with ωrule containing ID1. A derivation d in this system is transformed into a derivation g(d) with Ωrule introduced by W. Buchholz in such a way that one cutelimination step performed on a derivation d corresponds under g to passing to a subderivation of a derivation g(d). The termination of reduction steps follows by transfinite induction on the height of g(d). 1
ON THE ROLE OF RAMSEY QUANTIFIERS IN FIRST ORDER ARITHMETIC1
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ..."
Abstract
 Add to MetaCart
JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The