## Definitions of finiteness based on order properties

### Volume 189 / 2006

#### 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.