Abstract
In the framework of finitely supported atomic sets, by using the notion of atomic cardinality and the T-finite support principle (a closure property for supports in some higher-order constructions), we present some finiteness properties of the finitely supported binary relations between infinite atomic sets. Of particular interest are finitely supported Dedekind-finite sets because they do not contain finitely supported, countably infinite subsets. We prove that the infinite sets ℘fs(Ak×Al), ℘fs(Ak×℘m(A)), ℘fs(℘n(A)×Ak) and ℘fs(℘n(A)×℘m(A)) do not contain uniformly supported infinite subsets. Moreover, the functions space ZAm does not contain a uniformly supported infinite subset whenever Z does not contain a uniformly supported infinite subset. All these sets are Dedekind-finite in the framework of finitely supported structures.
Subject
Physics and Astronomy (miscellaneous),General Mathematics,Chemistry (miscellaneous),Computer Science (miscellaneous)
Reference20 articles.
1. Handbook of Mathematical Logic,1977
2. The Axiom of Choice;Jech,1973
3. Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre
4. Über die Unäbhangigkeit des Auswahlsaxioms und Einiger seiner Folgerungen;Lindenbaum;Comptes Rendus Séances Société Sci. Lettres Vars.,1938
5. Finitely Supported Mathematics: An Introduction;Alexandru,2016