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Ordinal analysis without proofs
 Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman. Lecture Notes in Logic 15
, 2002
"... Abstract. An approach to ordinal analysis is presented which is finitary, but highlights the semantic content of the theories under consideration, rather than the syntactic structure of their proofs. In this paper the methods are applied to the analysis of theories extending Peano arithmetic with tr ..."
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Abstract. An approach to ordinal analysis is presented which is finitary, but highlights the semantic content of the theories under consideration, rather than the syntactic structure of their proofs. In this paper the methods are applied to the analysis of theories extending Peano arithmetic with transfinite induction and transfinite arithmetic hierarchies. §1. Introduction. As the name implies, in the field of proof theory one tends to focus on proofs. Nowhere is this emphasis more evident than in the field of ordinal analysis, where one typically designs procedures for “unwinding” derivations in appropriate deductive systems. One might wonder, however, if this emphasis is really necessary; after all, the results of an ordinal
Dedicated to Wolfram Pohlers
"... On the limit existence principles in elementary ..."
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On Elementary Theories of Ordinal Notation Systems based on Reflection Principles
, 2013
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MATHEMATICAL COMMUNICATIONS 109 Math. Commun. 18(2013), 109–121 Π01ordinal analysis beyond firstorder arithmetic
, 2012
"... Abstract. In this paper we give an overview of an essential part of a Π01 ordinal analysis of Peano Arithmetic (PA) as presented by Beklemishev ([2]). This analysis is mainly performed within the polymodal provability logic GLPω. We reflect on ways of extending this analysis beyond PA. A main diffic ..."
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Abstract. In this paper we give an overview of an essential part of a Π01 ordinal analysis of Peano Arithmetic (PA) as presented by Beklemishev ([2]). This analysis is mainly performed within the polymodal provability logic GLPω. We reflect on ways of extending this analysis beyond PA. A main difficulty in this is to find proper generalizations of the socalled Reduction Property. The Reduction Property relates reflection principles to reflection rules. In this paper we prove a result that simplifies the reflection rules. Moreover, we see that for an ordinal analysis the full Reduction Property is not needed. This latter observation is also seen to open up ways for applications of ordinal analysis relative to some strong base theory.