Universics: a Theory of Universes of Discourse for Metamathematics and Foundations

Abstract

A new type of structures called ``universes'' is introduced to subsume the ``von Neumann universe'', ``Grothendieck universes'' and ``universes of discourse'' of various theories. Theories are also treated as universes, ``universes of ideas'', where ``idea" is a common term for assertions and terms. A dualism between induction and deduction and their treatment on a common basis is provided. The described approach referenced as ``universics'' is expected to be useful for metamathematical analysis and to serve as a foundation for mathematics. As a motivation for this research served the Harvey Friedman's desideratum to develop a foundational theory based on ``induction construction'', possibly comprising set theory. This desideratum emerged due to ``foundational incompleteness'' of set theory. The main results of this paper are an explication of the notion ``foundational completeness'', and a generalization of well-founded-ness.

Keywords

• induction
• deduction
• reduction
• universe