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IMCS/Publications/CSJM/Issues/CSJM v.24, n.1 (70), 2016/

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

Authors: Ioachim Drugus
Keywords: induction, deduction, reduction, universe

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.

Institute of Mathematics and Computer Science
Academy of Sciences of Moldova
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