Towards mathematical foundations of projects: Construction and formalization

This paper develops a formal axiomatic framework for Project Theory, introducing key concepts such as task dependencies, hierarchies, and project structures. The motivation behind this work is to achieve a global applicability by using universal mathematical language and logic, aiming to standardize the analysis of project management systems. We define projects in terms of well-founded relations and set-theoretic principles to rigorously describe their properties. Central to this theory are the Axiom of Compatibility, which ensures consistency across subprojects; the Axiom of Regularity, which prohibits infinite regress in project hierarchies and Mostowski?s Collapsing Theorem, which is applied to project structures to establish isomorphisms between transitive sets and projects. These theoretical results provide a robust foundation for modeling complex project management systems, enabling applications in areas such as operations research, workflow optimization, and software engineering.