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<cfResPublId>ibn-ResPubl-54540</cfResPublId>
<cfResPublDate>2015</cfResPublDate>
<cfVol>I</cfVol>
<cfStartPage>118</cfStartPage>
<cfISBN>978-9975-4237-3-1</cfISBN>
<cfURI>https://ibn.idsi.md/ro/vizualizare_articol/54540</cfURI>
<cfTitle cfLangCode='EN' cfTrans='o'>Universics: an Axiomatic Theory of Universes for the Foundations. Part 1. Foundational Completeness</cfTitle>
<cfKeyw cfLangCode='EN' cfTrans='o'>induction; deduction; reduction; well-foundedness; universics</cfKeyw>
<cfAbstr cfLangCode='EN' cfTrans='o'>This is the first part of a paper in 2 parts presenting an axiomatic theory called “universics” of mathematical structures called “universes” – structures like the “von Neumann universe”, “Grothendieck universes” and the “universes of discourse” of axiomatic theories. Universics is pivoted around a “reduction principle” expressed as an axiom scheme and manifesting both as a generalized epsilon-induction principle in set theory and, dually, as a principle of deduction in logic. The methodology of universics is to discuss about the reality in terms of “universes” treated a special kind of structures, rather than to discuss about it in terms of “theories”. As a motivation for this research served the Harvey Friedman’s desideratum to develop a new foundational theory richer than set theory, which would be essentially based on a generalized induction principle and into which, partially or completely, could be immersed set theory. This desideratum emerged due to the “foundational incompleteness” of set theory – a property manifesting as impossibility to represent in its language all mathematical structures and concepts. The main result of this Part 1 is an “explication”, i.e. a presentation in strict terms of universics, of the notion “foundational completeness”. In Part 2 an algebraic set theory based on the ideas of universics is developed, which is believed to achieve the Friedman’s desideratum. </cfAbstr>
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