Logic and Relativity

Madarasz J.X.

The subject matter of the dissertation is a mathematical logical investigation of the logical structure of (mainly special) relativity theories with an emphasis on the branch of logic called definability theory.Combining mathematical logic and relativity is not a new discipline, it goes back e.g. to Hilbert, Reichenbach, Carnap, Godel, Tarski, Suppes, Goldblatt. Definability theory, in particular, was initiated around f920 because of the needs of relativity theory. Then Tarski took a life-long interest in developing definability theory. In the dissertation we extend definability theory to the case when new elements and new universes of new elements can also be defined. We do this because we need it for studying relativity. We extend (and prove) the main theorems of definability from the classical case to the new situation. E.g. we prove a Beth style theorem about climinability of defined concepts. We apply these results to proving definitional equivalence of major, seemingly “disjoint” approaches to relativity. In particular, we prove a strong equivalence between purely geometrical theories and observation-oriented versions of relativity. Then we build so-called duality theories (on top of these equivalences) like adjoint situations in category theory. These duality theories connect various parts/versions of relativity with other, purely mathematical theories like Buseniann’s streamlined time-like-metric spaces.Besides the above, we strive to build up relativity as a theory purely in first-order logic using as simple and as transparent axioms as we can. One of our goals is to prove strong theorems of relativity from a small number of easily understandable, convincing axioms. We try to eliminate all tacit assumptions from relativity and replace them with explicit axioms in the spirit initiated by Tarski in his first-order axiomatization of geometry.