Definitions

# Finite set

In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where n is a natural number. (The value n = 0 is allowed; that is, the empty set is finite.) An infinite set is a set which is not finite.

Equivalently, a set is finite if its cardinality, i.e., the number of its elements, is a natural number. More specifically, a set whose cardinality is n is also called an n-set. For instance, the set of integers between −15 and 3 (excluding the end points) has 17 elements, so it is finite; in fact, it is a 17-set. In contrast, the set of all prime numbers has cardinality 0, so it is infinite.

A set is called Dedekind-finite if there exists no bijection between the set and any of its proper subsets. If the axiom of dependent choice (a weak form of the axiom of choice) holds, then a set is finite if and only if it is Dedekind-finite. Otherwise, paradoxically, there may be infinite Dedekind-finite sets (see Foundational issues below).

All finite sets are countable, but not all countable sets are finite. (However, some authors use "countable" to mean "countably infinite", and thus do not consider finite sets to be countable.)

## Closure properties

For any elements x, y, the sets {}, {x}, and {x, y} are finite. The union of a finite set of finite sets is finite. The powerset of a finite set is finite. Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite. The Cartesian product of a finite set of finite sets is finite. However, the set of natural numbers (whose existence is assured by the axiom of infinity) is not finite.

## Necessary and sufficient conditions for finiteness

In Zermelo–Fraenkel set theory (ZF), the following conditions are all equivalent:

1. S is a finite set. That is, S can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number.
2. (Kazimierz Kuratowski) S has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. (See the section on foundational issues for the set-theoretical formulation of Kuratowski finiteness.)
3. (Paul Stäckel) S can be given a total ordering which is both well-ordered forwards and backwards. That is, every non-empty subset of S has both a least and a greatest element in the subset.
4. Every function from P(P(S)) one-to-one into itself is onto. That is, the powerset of the powerset of S is Dedekind-finite (see below).
5. Every function from P(P(S)) onto itself is one-to-one.
6. (Alfred Tarski) Every non-empty family of subsets of S has a minimal element with respect to inclusion.
7. S can be well-ordered and any two well-orderings on it are order isomorphic. In other words, the well-orderings on S have exactly one order type.

If the axiom of choice also holds, then the following conditions are all equivalent:

1. S is a finite set.
2. (Richard Dedekind) Every function from S one-to-one into itself is onto.
3. Every function from S onto itself is one-to-one.
4. Every partial ordering of S contains a maximal element.

## Foundational issues

Georg Cantor initiated his theory of sets in order to provide a mathematical treatment of infinite sets. Thus the distinction between the finite and the infinite lies at the core of set theory. Certain foundationalists, the strict finitists, reject the existence of infinite sets and thus advocate a mathematics based solely on finite sets. Mainstream mathematicians consider strict finitism too confining, but acknowledge its relative consistency: the universe of hereditarily finite sets constitutes a model of Zermelo-Fraenkel set theory with the Axiom of Infinity replaced by its negation.

Even for those mathematicians who embrace infinite sets, in certain important contexts, the formal distinction between the finite and the infinite can remain a delicate matter. The difficulty stems from Gödel's incompleteness theorems. One can interpret the theory of hereditarily finite sets within Peano arithmetic (and certainly also vice-versa), so the incompleteness of the theory of Peano arithmetic implies that of the theory of hereditarily finite sets. In particular, there exists a plethora of so-called non-standard models of both theories. A seeming paradox, non-standard models of the theory of hereditarily finite sets contain infinite sets --- but these infinite sets look finite from within the model. (This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.) On account of the incompleteness theorems, no first-order predicate, nor even any recursive scheme of first-order predicates, can characterize the standard part of all such models. So, at least from the point of view of first-order logic, one can only hope to characterize finiteness approximately.

More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus. The best known axiomatic set theories include Zermelo-Fraenkel set theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), Von Neumann–Bernays–Gödel set theory (NBG), Non-well-founded set theory, Bertrand Russell's Type theory and all the theories of their various models. One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic.

A formalist might see the meaning of set varying from system to system. A Platonist might view particular formal systems as approximating an underlying reality.

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

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).

Call a set S Dedekind infinite if there exists an injective, non-surjective function $f:S rightarrow S$. Such a function exhibits a bijection between S and a proper subset of S, namely the image of f. Given an element x in a Dedekind infinite set S, we can form an infinite sequence of distinct elements of S, namely $x,f\left(x\right),f\left(f\left(x\right)\right),...$. Conversely, given a sequence in S consisting of elements $x_1,x_2,x_3,...$, we can define a function f such that on elements in the sequence $f\left(x_i\right)=x_\left\{i+1\right\}$ 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;
• X contains T implies X contains T union any singleton.

Let K(S) equal 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, as any infinite sequence of socks would effectively produce an impossible selection. But Kuratowski finiteness would fail for the same set of socks.