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Weak theories of nonstandard arithmetic and analysis
 Reverse Mathematics
, 2001
"... Abstract. A general method of interpreting weak highertype theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomialtime ..."
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Abstract. A general method of interpreting weak highertype theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomialtime computable arithmetic. A means of formalizing basic real analysis in such theories is sketched. §1. Introduction. Nonstandard analysis, as developed by Abraham Robinson, provides an elegant paradigm for the application of metamathematical ideas in mathematics. The idea is simple: use modeltheoretic methods to build rich extensions of a mathematical structure, like secondorder arithmetic or a universe of sets; reason about what is true in these enriched structures;
An Intuitionistic Logic That Proves Markov’s Principle
"... Abstract—We design an intuitionistic predicate logic that supports a limited amount of classical reasoning, just enough to prove a variant of Markov’s principle suited for predicate logic. At the computational level, the extraction of an existential witness out of a proof of its double negation is d ..."
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Abstract—We design an intuitionistic predicate logic that supports a limited amount of classical reasoning, just enough to prove a variant of Markov’s principle suited for predicate logic. At the computational level, the extraction of an existential witness out of a proof of its double negation is done by using a form of staticallybound exception mechanism, what can be seen as a directstyle variant of Friedman’s Atranslation. KeywordsMarkov’s principle; intuitionistic logic; proofasprogram correspondence; exceptions I.
An MRcomplete extension of extension of TRDB and its functional interpretation
"... In this paper, we extend a functional interpretation to transfinite types. To be more precise, we define a modified realizability interpretation of a constructive arithmetic with a certain inductive definition and barinduction. The system is called TRDB. We also give an extension of TRDB, which is ..."
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In this paper, we extend a functional interpretation to transfinite types. To be more precise, we define a modified realizability interpretation of a constructive arithmetic with a certain inductive definition and barinduction. The system is called TRDB. We also give an extension of TRDB, which is called S, and show that S is a complete system with respect to the modified realizability interpretation. In the second half, we define a truthvalue of Sformulas under certain functionals, and show validity of S with respect to the truthvalue. 1 Introduction In a previous work [5], Yasugi and Hayashi formulated a system of constructive arithmetic with transfinite recursion and bar induction. This system was called TRDB, which is a streamlined version of the system used by Yasugi in [4] to prove the accessibility of an order system. TRDB is, however, a mathematically interesting system on its own right. For this reason, it has been studied from various aspects. [5] reformulates the ar...
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)