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'. Ordinal-theoretic 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...

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 statically-bound exception mechanism, what can be seen as a direct-style variant of Friedman’s A-translation. Keywords-Markov’s principle; intuitionistic logic; proof-asprogram correspondence; exceptions I.

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 bar-induction. 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 truth-value of S-formulas under certain functionals, and show validity of S with respect to the truth-value. 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] re-formulates the ar...

Abstract. In this paper we introduce a notation system for the infinitary derivations occurring in the ordinal analysis of KP + Π3-Reflection due to Michael Rathjen. This allows a finitary ordinal analysis of KP+Π3-Reflection. 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+Π3-Reflection as <-recursive functions where < is the ordering on Rathjen’s ordinal notation system T (K). Further we show a conservation result for Π 0 2-sentences. §1. Introduction. Ordinal analysis uses cut-elimination techniques for proof theoretic investigations. The termination of the cut-elimination 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)

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