Finite Set - Set-theoretic Definitions of Finiteness

Set-theoretic Definitions of Finiteness

In contexts where the notion of natural number sits logically prior to any notion of set, one can define a set S as finite if S admits a bijection to some set of natural numbers of the form . Mathematicians more typically choose to ground notions of number in set theory, for example they might model natural numbers by the order types of finite well-ordered sets. Such an approach requires a structural definition of finiteness that does not depend on natural numbers.

Interestingly, various properties that single out the finite sets among all sets in the theory ZFC turn out logically inequivalent in weaker systems such as ZF or intuitionistic set theories. Two definitions feature prominently in the literature, one due to Richard Dedekind, the other to Kazimierz Kuratowski (Kuratowski's is the definition used above).

A set S is called Dedekind infinite if there exists an injective, non-surjective function . Such a function exhibits a bijection between S and a proper subset of S, namely the image of f. Given a Dedekind infinite set S, a function f, and an element x that is not in the image of f, we can form an infinite sequence of distinct elements of S, namely . Conversely, given a sequence in S consisting of distinct elements, we can define a function f such that on elements in the sequence and f behaves like the identity function otherwise. Thus Dedekind infinite sets contain subsets that correspond bijectively with the natural numbers. Dedekind finite naturally means that every injective self-map is also surjective.

Kuratowski finiteness is defined as follows. Given any set S, the binary operation of union endows the powerset P(S) with the structure of a semi-lattice. Writing K(S) for the sub-semi-lattice generated by the empty set and the singletons, call set S Kuratowski finite if S itself belongs to K(S). Intuitively, K(S) consists of the finite subsets of S. Crucially, one does not need induction, recursion or a definition of natural numbers to define generated by since one may obtain K(S) simply by taking the intersection of all sub-semi-lattices containing the empty set and the singletons.

Readers unfamiliar with semi-lattices and other notions of abstract algebra may prefer an entirely elementary formulation. Kuratowski finite means S lies in the set K(S), constructed as follows. Write M for the set of all subsets X of P(S) such that:

• X contains the empty set;
• For every set T in P(S), if X contains T then X also contains the union of T with any singleton.

Then K(S) may be defined as the intersection of M.

In ZF, Kuratowski finite implies Dedekind finite, but not vice-versa. In the parlance of a popular pedagogical formulation, when the axiom of choice fails badly, one may have an infinite family of socks with no way to choose one sock from more than finitely many of the pairs. That would make the set of such socks Dedekind finite: there can be no infinite sequence of socks, because such a sequence would allow a choice of one sock for infinitely many pairs by choosing the first sock in the sequence. However, Kuratowski finiteness would fail for the same set of socks.