Algebrai logika; relativitáselmélet logikai struktúrájának vizsgálata = Algebraic logic; investigating the logical structure of relativity theory

Gödel, Einstein és Tarski hagyományait kívánjuk folytatni, elmélyítve a Gödel-Einstein együttműködés eredményeit is, és folytatva Tarski tudományegyesítési programmját. Ismert, hogy a logika és a matematika modern megalapozása Gödel és Tarski úttörő munkásságára vezethető vissza. Kevésbbé ismert, hogy Gödel 1948-tól majdnem élete végéig Einsteinnel szorosan együttműködve relativitáselméleten dolgozott, ahol ugyanolyan meghökkentő új horizontokat tárt fel mint logikában, és hogy Gödel relativitáselméleti gondolatai folytatásaként fogható fel a forgó fekete lyukak mai elmélete. Ezen előzmények folytatása a jelen projektum, mely Tarskival és munkatársaival való személyes együttműködés (pl. közös könyv) keretében kezdődött. Az alapgondolat a logika, algebra, geometria, téridőelmélet és relativitáselmélet egységben való művelése. Eredményeinkből egy példa: Nagy, lassan forgó fekete lyukakról bizonyítottuk, hogy a belsejében létrejövő un. zárt időszerű görbe (időhurok) létrejöttére vonatkozó szokásos irodalmi magyarázatok tévesek. Nem az un. drag effect (mozgó anyag magával vonszolja a téridőt) okozza a zárt görbéket, hanem egy egészen más jellegű hatás: a fénykúpok kinyílása a forgással ellentétes irányban. Az eredmény a General Relativity and Gravitation című folyóiratban jelenik meg. | The reported project intends to continue traditions of Gödel, Einstein and Tarski continuing the spirit of the Gödel-Einstein collaboration and pursuing Tarski's programme for unifying science. Modern logic and meta-mathematics was created (basically) by Gödel and Tarski. It is less well known that beginning with 1948 Gödel spent much time with Einstein and worked on relativity theory. Of course, he remained a logician in spirit. Gödel obtained fundamental breakthroughs in relativity like his ones in logic and foundations. The theory of general relativistic spacetimes not admitting a global Time was initiated by Gödel, and came to full blossom during the renaissance of black hole physics during the last 25 years. The present project was originally started in personal cooperation with Tarski and his collaborators. The idea is to study logic, algebra, geometry, spacetime theory and relativity in a strong unity. A sample result of ours: We proved about big, slowly rotating black holes that the usual explanation in the literature of why such black holes contain a closed timelike curve (CTC) is flawed. Namely, it is not the gravitational frame dragging effect which creates CTCs, instead, there is a completely different kind of effect in action there: light cones open up in the direction opposite to that of the rotation of the source and this goes on to such an extreme extent that CTCs are created. Our paper on this appears in the journal General Relativity and Gravitation

Complexity of equational theory of relational algebras with standard projection elements

The class $\mathsf{TPA}$ of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of $\mathsf{TPA}$ nor the first order theory of $\mathsf{TPA}$ are decidable. Moreover, we show that the set of all equations valid in $\mathsf{TPA}$ is exactly on the $\Pi ^1_1$ level. We consider the class $\mathsf{TPA}^-$ of the relation algebra reducts of $\mathsf{TPA}$’s, as well. We prove that the equational theory of $\mathsf{TPA}^-$ is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work

Weak products of universal algebras

Weak direct products of arbitrary universal algebras are introduced. The usual notion for groups and rings is a special case. Some universal algebraic properties are proved and applications to cylindric and polyadic algebras are considered

Temporal logics need their clocks

AbstractWe investigate effective inference systems for first-order temporal logics from the point of view of completeness and soundness. Among others, the role of clocks in these issues will be somewhat clarified by our results. Some open problems from the literature of temporal logic will be solved

The life and work of Leon Henkin: essays on his contributions

This is a comprehensive book on the life and works of Leon Henkin (1921–2006), an extraordinary scientist and excellent teacher whose writings became influential right from the beginning of his career with his doctoral thesis on “The completeness of formal systems” under the direction of Alonzo Church. Upon the invitation of Alfred Tarski, Henkin joined the Group in Logic and the Methodology of Science in the Department of Mathematics at the University of California Berkeley in 1953. He stayed with the group until his retirement in 1991. This edited volume includes both foundational material and a logic perspective. Algebraic logic, model theory, type theory, completeness theorems, philosophical and foundational studies are among the topics covered, as well as mathematical education. The work discusses Henkin’s intellectual development, his relation to his predecessors and contemporaries, and his impact on the recent development of mathematical logic. It offers a valuable reference work for researchers and students in the fields of philosophy, mathematics and computer science