On foundational frames for formal modelling

Any formal modelling is influenced by a particular view of the world - or part of it - which is present in the foundational frame in which the modelling is done. This work deals with such foundational frames which are fundamental in formal sciences: frames which present a view on the world of sets. The work splits into two parts. In the first part the foundational frames taken into account are introduced in order to examine their strength for set-theoretic modelling. They are presented in a model-theoretical Tarski-Style using the notion of a universe. The set theory of Zermelo-Fraenkel together with the Axiom of Choice, ZFC, is a widely accepted foundational frame which is presented by the Universe V of all sets for which the ZFC-Axioms are assumed to be true. It is possible to develop classical mathematics in V but ZFC holds some regularities and restrictions which make formal modelling of reflexive and intensional structures in computer science and linguistics difficult. Instead of replacing ZFC by an alternative set theory as a foundational frame for such modelling, a foundational frame within the frame of V is modelled in which these restrictions are offset. This is done by developing the theory of ɛ-structures and ɛ-style semantics in V. The new frame of ɛ-sets, although being modelled in V, is in fact a generalization of ZFC: the universe Uɛ of ɛ-sets allows to naturally embed the ZFC-frame V into it. Moreover, among the restrictions which are offset within the frame of ɛ-sets, there is the Axiom of Foundation. The theory of non-well founded sets of Aczel is an alternative set-theory, developed by Aczel to avoid Foundation. It turns out that Aczel’s theory can be modelled in the frame of ɛ-sets by factorization. Hence both, ZFC and Aczel’s theory are represented in that frame. These results do not only show the modelling strength of ɛ-style semantics but at the same time also underline the modelling strength of ZFC.