. After a brief flirtation with logicism in 1917--1920, 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 e-calculus, on which Hilbert's axiom systems were based, and the development of the e-substitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the e-notation and provides an analysis of Ackermann's consisten...

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 1-comprehension rule.

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 Martin-Lof type theory with W-type and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of Martin-Lof type theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis. Finally we repeat the concept of inductive-recursive 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.

We apply Mints ’ technique for proving the termination of the epsilon substitution method via cut-elimination to the system of Peano Arithmetic with Transfinite Induction given by Arai. 1

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

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 inimene-masin kui ka arvutisisesel tasandil), tuletusmeetodid ja "teadmiste spetsifitseerimise tehnoloogia" (teadmustehnika) . 1993.a. uuringute tulemused h~olmavad jargmisi valdkondi. ffl Teadmiste ja andmete objekt-orienteeritud 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 ...