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First order theories for nonmonotone inductive definitions: recursively inaccessible and Mahlo
, 1998
"... In this paper first order theories for nonmonotone inductive definitions are introduced, and a prooftheoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given. 1 Introduction Let # be an operator on the power ..."
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In this paper first order theories for nonmonotone inductive definitions are introduced, and a prooftheoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given. 1 Introduction Let # be an operator on the power set P (N) of the natural numbers, i.e. a mapping from P (N) to P (N). Then # can be used to generate subsets I # # of the natural numbers if we define I # # := I <# # # #(I <# # ) and I <# # := # {I # # : # < #} by transfinite recursion on the ordinals. Furthermore we let I # # := # {I # # : # an ordinal } be the set of natural numbers inductively defined by #. Obviously there exists a least ordinal # so that I # # = I <# # . We call this ordinal the closure ordinal of the inductive definition generated by # and know that I # # is identical to I # # . The sets I # # are the stages of the inductive definition generated by #. If K is a class of operators, th...
Operations, sets and classes
"... Operational set theory, in the form described below, is an enterprise which consolidates classical set theory with some central concepts of Feferman’s explicit mathematics. It provides for a careful distinction between operations and settheoretic functions and as such reconciles set theory with nee ..."
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Operational set theory, in the form described below, is an enterprise which consolidates classical set theory with some central concepts of Feferman’s explicit mathematics. It provides for a careful distinction between operations and settheoretic functions and as such reconciles set theory with needs arising in constructive environments and even in those enhanced by computer science. In the following we consider, primarily from a prooftheoretic perspective, the theory OST and some of its most important extensions and determine their consistency strengths by exhibiting equivalent systems in the realm of traditional theories of sets and classes.
Finitary reductions for local predicativity, I: recursively regular ordinals
"... We define notation system for infinitary derivations arising from cutelimination for a theory T 1 \Sigma of recursively regular ordinals by the method of local predicativity. Using these notations, we derive finitary cutelimination steps together with corresponding ordinal assignments. Introductio ..."
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We define notation system for infinitary derivations arising from cutelimination for a theory T 1 \Sigma of recursively regular ordinals by the method of local predicativity. Using these notations, we derive finitary cutelimination steps together with corresponding ordinal assignments. Introduction There is an extensive literature connecting infinitary "Schuttestyle" and finitary "GentzenTakeutistyle" sides of proof theory. For example, in papers [Mi75, Mi75a, Mi79, Bu91, Bu97a] this was done for systems not exceeding in strength Peano Arithmetic. But most recently, there has been an interest to what one can get on the side of finitary proof theory from the methods which are used for prooftheoretical analysis of impredicative theories (see [Wei96, Bu97]). Especially we want to mention paper [Bu97], where it was shown that Takeuti's reduction steps for \Pi 1 1 \Gamma CA+ BI [Tak87, x27] can be derived from Buchholz' method of\Omega +1 rule ([BFPS, Ch. IVV], [BS88]). Here we ...
A BUCHOLZ DERIVATION SYSTEM FOR THE ORDINAL ANALYSIS OF KP + Π3REFLECTION
"... Abstract. In this paper we introduce a notation system for the infinitary derivations occurring in the ordinal analysis of KP + Π3Reflection due to Michael Rathjen. This allows a finitary ordinal analysis of KP+Π3Reflection. The method used is an extension of techniques developed by Wilfried Buchh ..."
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Abstract. In this paper we introduce a notation system for the infinitary derivations occurring in the ordinal analysis of KP + Π3Reflection due to Michael Rathjen. This allows a finitary ordinal analysis of KP+Π3Reflection. The method used is an extension of techniques developed by Wilfried Buchholz, namely operator controlled notation systems for RS ∞derivations. Similarly to Buchholz we obtain a characterisation of the provably recursive functions of KP+Π3Reflection as <recursive functions where < is the ordering on Rathjen’s ordinal notation system T (K). Further we show a conservation result for Π 0 2sentences. §1. Introduction. Ordinal analysis uses cutelimination techniques for proof theoretic investigations. The termination of the cutelimination process is guaranteed by assigning decreasing ordinals to the proofs emerging in the process. Gerhard Gentzen was the first to form a relationship between an ordinal ε0 and a foundational mathematical theory (nowadays denoted Peano Arithmetic PA)
Contents
, 2013
"... • A brief history of proof theory • Sequent calculi for classical and intuitionistic logic, Gentzen’s Hauptsatz: Cut elimination ..."
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• A brief history of proof theory • Sequent calculi for classical and intuitionistic logic, Gentzen’s Hauptsatz: Cut elimination