Definitions of finiteness based on order properties

Volume 189 / 2006

Omar De la Cruz, Damir D. Dzhafarov, Eric J. Hall Fundamenta Mathematicae 189 (2006), 155-172 MSC: 03E25, 03E20, 03E35, 06A07. DOI: 10.4064/fm189-2-5

Abstract

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.

Authors

  • Omar De la CruzDepartment 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.
    e-mail
  • Damir D. DzhafarovDepartment 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.
    e-mail
  • Eric J. HallDepartment of Mathematics and Statistics
    University of Missouri–Kansas City
    5100 Rockhill Rd.
    Kansas City, MO 64110-2499, U.S.A.
    e-mail

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