According to Suppes [p. 100], Tarski proposed a slick definition of finiteness which mentions neither the natural numbers nor bijections. Here it is (quoting Suppes):

``A is finite if and only if every non-empty family of subsets of A has a minimal element. ''

In this context, a minimal element in a family F of subsets is an element x in F such that there is no y in F such that y is a *proper* subset of x.

Granted, at first glace, this definition is not intuitively clear to most people (i.e. non set-theorists). But one can probably get used to it after a while. This is the formal definition that Suppes adopts in his text.

Here's a cool fact: Unlike Dedekind's definition (i.e. a set is finite if and only if it is not bijective to a proper subset), ``Tarski's definition does not require the axiom of choice to prove its equivalence to the ordinary numerical definition." [Suppes, p. 99]

According to Suppes, Tarski proposed this definition in his paper
``Sur les ensembles finis," published in _Fundamenta_Mathematicae_, Vol 6 (1924b), pp. 45 - 95. This would be an interesting paper to read. Not only does Tarski propose a new definition, he also gives a complete survey of all of the ``non-numerical definitions of finitude" up to that time.

Reference:

Suppes, P. _Axiomatic_Set_Theory_, Dover, New York: 1972