Formalizing Mathematics via Predicative and Constructive Approaches

Formalized mathematics and mathematical knowledge management (MKM) are extremely fruitful and quickly expanding fields of research at the intersection of mathematics and computer science. The declared goal of these fields is to develop computerized systems that effectively represent all important mathematical knowledge and techniques, while conforming to the highest standards of mathematical rigor. The use of proof assistants or interactive theorem provers has seen tremendous growth in recent years because of its record in assisting with the development and verification of formal proofs by human-machine collaboration. It has also found numerous high value applications in different research areas, including cyber security, cyber physical systems, correct-by-construction programming, and advanced programming language design. This rapidly developing field is bound to ultimately have a huge impact on the culture of mathematical practice and education. Recent years, however, have seen an estrangement between the informal mathematical practice and the mainstream work in the field of MKM. Thus, while set theory is viewed by most mathematicians as the foundation of the mathematics they practice, this is not reflected in most extant proof assistants. We believe that the use for MKM of a set-theoretical frameworks could help overcome this increasing rift between what most mathematicians consider to be the basis of mathematics, and what existing formal-reasoning systems actually implement. Accordingly, this thesis aims at a formalization of applicable mathematics on the basis of a convenient formal set-theoretical framework which is suitable for mechanization and reflects real mathematical practice as presented in ordinary (or computationally-oriented) mathematical discourse. Another related goal of this thesis, motivated also by philosophical considerations, is to identify the minimal ontological commitments required for the above-mentioned task of the formalization of applicable mathematics. These (different, but related) goals can be simultaneously pursued by exploiting the modularity of the framework employed, which enables the use of different logics and set theories of different strength. Therefore, in this work we consider the following variations of the framework and of the theories developed within it: • The underlying logic can be classical logic, or, in cases where the computational power of a theory should be enhanced, intuitionistic (constructive) logic.