Definitions of finiteness based on order properties

Omar De la Cruz ¹   ; Damir D. Dzhafarov ²   ; Eric J. Hall ³

¹ Department of Mathematics Purdue University West Lafayette, IN, U.S.A. and Centre de Recerca Matematica Barcelona, Spain and Department of Statistics University of Chicago 5734 S. University Avenue Chicago, Illinois 60637-1514, U.S.A.
² Department of Mathematics Purdue University West Lafayette, IN, U.S.A. and Department of Mathematics University of Chicago 5734 S. University Avenue Chicago, Illinois 60637-1514, U.S.A.
³ Department of Mathematics and Statistics University of Missouri–Kansas City 5100 Rockhill Rd. Kansas City, MO 64110-2499, U.S.A.

Fundamenta Mathematicae, Tome 189 (2006) no. 2, pp. 155-172

Résumé

A definition of finiteness is a set-theoretical property of a set that, if the Axiom of Choice (AC) is assumed, is equivalent to stating that the set is finite; several such definitions have been studied over the years. In this article we introduce a framework for generating definitions of finiteness in a systematical way: basic definitions are obtained from properties of certain classes of binary relations, and further definitions are obtained from the basic ones by closing them under subsets or under quotients.We work in set theory without AC to establish relations of implication and independence between these definitions, as well as between them and other notions of finiteness previously studied in the literature. It turns out that several well known definitions of finiteness (including Dedekind finiteness) fit into our framework by being equivalent to one of our definitions; however, a few of our definitions are actually new. We also show that Ia-finite unions of Ia-finite sets are P-finite (one of our new definitions), but that the class of P-finite sets is not provably closed under unions.

Détail

DOI : 10.4064/fm189-2-5

Keywords: definition finiteness set theoretical property set axiom choice assumed equivalent stating set finite several definitions have studied years article introduce framework generating definitions finiteness systematical basic definitions obtained properties certain classes binary relations further definitions obtained basic closing under subsets under quotients work set theory without establish relations implication independence between these definitions between other notions finiteness previously studied literature turns out several known definitions finiteness including dedekind finiteness fit framework being equivalent definitions however few definitions actually ia finite unions ia finite sets p finite definitions class p finite sets provably closed under unions