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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, a finite set is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :$\$ is a finite set with five elements. The number of elements of a finite set is a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
(possibly zero) and is called the ''
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
(or the
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: :$\.$ Finite sets are particularly important in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
function from a larger finite set to a smaller finite set.

# Definition and terminology

Formally, a set is called finite if there exists a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
:$f\colon S\to\$ for some natural number . The number is the set's cardinality, denoted as . The
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
or ∅ is considered finite, with cardinality zero. If a set is finite, its elements may be written — in many ways — in a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
: :$x_1,x_2,\ldots,x_n \quad \left(x_i \in S, \ 1 \le i \le n\right).$ In
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
, a finite set with elements is sometimes called an ''-set'' and a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
with elements is called a ''-subset''. For example, the set is a 3-set – a finite set with three elements – and is a 2-subset of it. (Those familiar with the definition of the natural numbers themselves as conventional in
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, the so-called von Neumann construction, may prefer to use the existence of the bijection $f \colon S \to n$, which is equivalent.)

# Basic properties

Any
proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of a finite set ''S'' is finite and has fewer elements than ''S'' itself. As a consequence, there cannot exist a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between a finite set ''S'' and a proper subset of ''S''. Any set with this property is called Dedekind-finite. Using the standard ZFC axioms for
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, every Dedekind-finite set is also finite, but this implication cannot be proved in ZF (Zermelo–Fraenkel axioms without the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
) alone. The axiom of countable choice, a weak version of the axiom of choice, is sufficient to prove this equivalence. Any injective function between two finite sets of the same cardinality is also a surjective function (a surjection). Similarly, any surjection between two finite sets of the same cardinality is also an injection. The union of two finite sets is finite, with :$, S \cup T, \le , S, + , T, .$ In fact, by the inclusion–exclusion principle: :$, S \cup T, = , S, + , T, - , S\cap T, .$ More generally, the union of any finite number of finite sets is finite. The
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
of finite sets is also finite, with: :$, S \times T, = , S, \times, T, .$ Similarly, the Cartesian product of finitely many finite sets is finite. A finite set with ''n'' elements has 2 distinct subsets. That is, the power set ''P''(''S'') of a finite set ''S'' is finite, with cardinality 2. 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. All finite sets are
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do not consider finite sets to be countable.) The
free semilattice In mathematics, in the area of order theory, a free lattice is the free object corresponding to a Lattice (order), lattice. As free objects, they have the universal property. Formal definition Any set (mathematics), set ''X'' may be used to generat ...
over a finite set is the set of its non-empty subsets, with the join operation being given by set union.

# Necessary and sufficient conditions for finiteness

In
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
without the axiom of choice (ZF), the following conditions are all equivalent: # ''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. # ( 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
below Below may refer to: *Earth * Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Ernst von Below (1863–1955), German World War I general *Fred Below ...
for the set-theoretical formulation of Kuratowski finiteness.) # (
Paul Stäckel Paul Gustav Samuel Stäckel (20 August 1862, Berlin – 12 December 1919, Heidelberg) was a German mathematician, active in the areas of differential geometry, number theory, and non-Euclidean geometry. In the area of prime number theory, h ...
) ''S'' can be given a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
ing which is
well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-or ...
ed both forwards and backwards. That is, every non-empty subset of ''S'' has both a least and a greatest element in the subset. # Every one-to-one function from ''P''(''P''(''S'')) into itself is onto. That is, the
powerset In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of the powerset of ''S'' is Dedekind-finite (see below). # Every surjective function from ''P''(''P''(''S'')) onto itself is one-to-one. # ( Alfred Tarski) Every non-empty family of subsets of ''S'' has a minimal element with respect to inclusion. (Equivalently, every non-empty family of subsets of ''S'' has a
maximal element In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defin ...
with respect to inclusion.) # ''S'' can be well-ordered and any two well-orderings on it are
order isomorphic In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be ...
. In other words, the well-orderings on ''S'' have exactly one
order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y suc ...
. If the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
is also assumed (the axiom of countable choice is sufficient), then the following conditions are all equivalent: # ''S'' is a finite set. # (
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
) Every one-to-one function from ''S'' into itself is onto. # Every surjective function from ''S'' onto itself is one-to-one. # ''S'' is empty or every partial ordering of ''S'' contains a
maximal element In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defin ...
.

# Foundational issues

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
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 recommend 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 In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
with the axiom of infinity replaced by its
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
. Even for the majority of mathematicians that 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 Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phil ...
. One can interpret the theory of hereditarily finite sets within
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearl ...
(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 is that there are non-standard models of the theory of hereditarily finite sets which 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 In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hig ...
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 describe finiteness approximately. More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A fo ...
s 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 Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
's
Type theory In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...
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. Some kinds of Platonists might view particular formal systems as approximating an underlying reality.

# Set-theoretic definitions of finiteness

In contexts where the notion of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
sits logically prior to any notion of set, one can define a set ''S'' as finite if ''S'' admits a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
to some set of natural numbers of the form $\$. Mathematicians more typically choose to ground notions of number in
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, 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. 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 Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
, 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 $f:S \rightarrow S$. 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 $x,f\left(x\right),f\left(f\left(x\right)\right),...$. Conversely, given a sequence in ''S'' consisting of distinct elements $x_1, x_2, x_3, ...$, we can define a function ''f'' such that on elements in the sequence $f\left(x_i\right) = x_$ 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 In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
''P''(''S'') with the structure of a semilattice. Writing ''K''(''S'') for the sub-semilattice 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-semilattices containing the empty set and the singletons. Readers unfamiliar with semilattices 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.

## Other concepts of finiteness

In ZF set theory without the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, the following concepts of finiteness for a set ''S'' are distinct. They are arranged in strictly decreasing order of strength, i.e. if a set ''S'' meets a criterion in the list then it meets all of the following criteria. In the absence of the axiom of choice the reverse implications are all unprovable, but if the axiom of choice is assumed then all of these concepts are equivalent. (Note that none of these definitions need the set of finite
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s to be defined first; they are all pure "set-theoretic" definitions in terms of the equality and membership relations, not involving ω.) * I-finite. Every non-empty set of subsets of ''S'' has a ⊆-maximal element. (This is equivalent to requiring the existence of a ⊆-minimal element. It is also equivalent to the standard numerical concept of finiteness.) * Ia-finite. For every partition of ''S'' into two sets, at least one of the two sets is I-finite. (A set with this property which is not I-finite is called an
amorphous set In set theory, an amorphous set is an infinite set which is not the disjoint union of two infinite subsets.. Existence Amorphous sets cannot exist if the axiom of choice is assumed. Fraenkel constructed a permutation model of Zermelo–Fraen ...
.) * II-finite. Every non-empty ⊆-monotone set of subsets of ''S'' has a ⊆-maximal element. * III-finite. The power set ''P''(''S'') is Dedekind finite. * IV-finite. ''S'' is Dedekind finite. * V-finite. ∣''S''∣ = 0 or 2 ⋅ ∣''S''∣ > ∣''S'', . * VI-finite. ∣''S''∣ = 0 or ∣''S''∣ = 1 or ∣''S''∣2 > ∣''S''∣. * VII-finite. ''S'' is I-finite or not well-orderable. The forward implications (from strong to weak) are theorems within ZF. Counter-examples to the reverse implications (from weak to strong) in ZF with
urelement In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Theory There ...
s are found using
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
. found counter-examples to each of the reverse implications in Mostowski models. Lévy attributes most of the results to earlier papers by Mostowski and Lindenbaum. Most of these finiteness definitions and their names are attributed to by . However, definitions I, II, III, IV and V were presented in , together with proofs (or references to proofs) for the forward implications. At that time, model theory was not sufficiently advanced to find the counter-examples. Each of the properties I-finite thru IV-finite is a notion of smallness in the sense that any subset of a set with such a property will also have the property. This is not true for V-finite thru VII-finite because they may have countably infinite subsets.

* FinSet *
Ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
*
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearl ...

# References

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