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The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program
, 2001
"... . After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the ..."
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. After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermann 's dissertation of 1924. The main innovation was the invention of the ecalculus, on which Hilbert's axiom systems were based, and the development of the esubstitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the enotation and provides an analysis of Ackermann's consisten...
Epsilonsubstitution method for the ramified language and # 1 comprehension rule
 Logic and Foundations of Mathematics
, 1999
"... We extend to Ramified Analysis the definition and termination proof of Hilbert’s ɛsubstitution method. This forms a base for future extensions to predicatively reducible subsystems of analysis. First such system treated here is second order arithmetic with ∆1 1comprehension rule. ..."
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We extend to Ramified Analysis the definition and termination proof of Hilbert’s ɛsubstitution method. This forms a base for future extensions to predicatively reducible subsystems of analysis. First such system treated here is second order arithmetic with ∆1 1comprehension rule.
Proof Theory of MartinLof Type Theory  An
 Mathematiques et Sciences Humaines, 42 année, n o 165:59 – 99
, 2004
"... We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsisten ..."
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We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of MartinLof type theory with Wtype and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of MartinLof type theory with Wtype and a universe closed under the Wtype, and consider the extension of type theory by one Mahlo universe and its prooftheoretic analysis. Finally we repeat the concept of inductiverecursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength.
Epsilon substitution for transfinite induction
 Arch. Math. Logic
, 2005
"... We apply Mints ’ technique for proving the termination of the epsilon substitution method via cutelimination to the system of Peano Arithmetic with Transfinite Induction given by Arai. 1 ..."
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We apply Mints ’ technique for proving the termination of the epsilon substitution method via cutelimination to the system of Peano Arithmetic with Transfinite Induction given by Arai. 1
Cut Elimination for a Simple Formulation of Epsilon Calculus
, 2007
"... A simple cut elimination proof for arithmetic with epsilon symbol is used to establish termination of a modified epsilon substitution process. This opens a possibility of extension to much stronger systems. 1 ..."
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A simple cut elimination proof for arithmetic with epsilon symbol is used to establish termination of a modified epsilon substitution process. This opens a possibility of extension to much stronger systems. 1
Languages, Tools and Methods for Conceptual Modelling
, 1993
"... Kontseptuaalse modelleerimise keeled, vahendid ja meetodid Aruandes kasitletakse Kuberneetika Instituudi tarkvaraosakonna 1993.a. uurimistulemusi. Vaadeldava uurimistoo sisuks on teadmiste esitamise ja kasutamise mudelite taiustamine eesmargiga valja arendada teadmiste baasidega koos tootava progr ..."
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Kontseptuaalse modelleerimise keeled, vahendid ja meetodid Aruandes kasitletakse Kuberneetika Instituudi tarkvaraosakonna 1993.a. uurimistulemusi. Vaadeldava uurimistoo sisuks on teadmiste esitamise ja kasutamise mudelite taiustamine eesmargiga valja arendada teadmiste baasidega koos tootava programmeerimiskeskkonna alused, kaasaarvatud erinevat tuupi teadmiste esitamise mudelid ja keeled (nii inimenemasin kui ka arvutisisesel tasandil), tuletusmeetodid ja "teadmiste spetsifitseerimise tehnoloogia" (teadmustehnika) . 1993.a. uuringute tulemused h~olmavad jargmisi valdkondi. ffl Teadmiste ja andmete objektorienteeritud mudelite taiustamine. Objektorienteeritud susteemide ja andmebaaside dunaamiliste aspektide modelleerimiseks vajalike kirjeldus, teisendus ja tuletusmeetodite valjatootamine (M.Matskin, H.M.Haav, S.Tupailo). ffl Lihtsate atribuutautomaatide komponeerimise/dekomponeerimise algebralise teooria valjatootamine ning atribuutautomaadi teatud aspektide modelleerimine ...
Realizability and Strong Normalization for a CurryHoward Interpretation of HA + EM1
"... We present a new CurryHoward correspondence for HA + EM1, constructive Heyting Arithmetic with the excluded middle on Σ 0 1formulas. We add to the lambda calculus an operator ‖a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoin ..."
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We present a new CurryHoward correspondence for HA + EM1, constructive Heyting Arithmetic with the excluded middle on Σ 0 1formulas. We add to the lambda calculus an operator ‖a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoint of logic, a restricted version of the excluded middle. We motivate the restriction of the excluded middle by its use in proof mining; we introduce new techniques to prove strong normalization for HA + EM1 and the witness property for simply existential statements. One may consider our results as an application of the ideas of Interactive realizability, which we have adapted to the new setting and used to prove our main theorems.
Interactive Realizability for Classical Peano Arithmetic with Skolem Axioms
"... Interactive realizability is a computational semantics of classical Arithmetic. It is based on interactive learning and was originally designed to interpret excluded middle and Skolem axioms for simple existential formulas. A realizer represents a proof/construction depending on some state, which is ..."
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Interactive realizability is a computational semantics of classical Arithmetic. It is based on interactive learning and was originally designed to interpret excluded middle and Skolem axioms for simple existential formulas. A realizer represents a proof/construction depending on some state, which is an approximation of some Skolem functions. The realizer interacts with the environment, which may provide a counterproof, a counterexample invalidating the current construction of the realizer. But the realizer is always able to turn such a negative outcome into a positive information, which consists in some new piece of knowledge learned about the mentioned Skolem functions. The aim of this work is to extend Interactive realizability to a system which includes classical firstorder Peano Arithmetic with Skolem axioms. For witness extraction, the learning capabilities of realizers will be exploited according to the paradigm of learning by levels. In particular, realizers of atomic formulas will be update procedures in the sense of Avigad and thus will be understood as stratifiedlearning algorithms.