In

^{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 urelements are found using

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, a finite set is a set 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,
:$\backslash $
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 n ...

(possibly zero) and is called the '' cardinality (or the cardinal number)'' 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:
:$\backslash .$
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 set, finite Mathematical structure, structures. It is closely related to many other ar ...

, the mathematical study of counting
Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...

. Many arguments involving finite sets rely on the pigeonhole principle
In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there mu ...

, which states that there cannot exist an injective function from a larger finite set to a smaller finite set.
Definition and terminology

Formally, a set is called finite if there exists abijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...

:$f\backslash colon\; S\backslash to\backslash $
for some natural number . The number is the set's cardinality, denoted as . The empty set
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 ...

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 mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...

:
:$x\_1,x\_2,\backslash ldots,x\_n\; \backslash quad\; (x\_i\; \backslash in\; S,\; \backslash \; 1\; \backslash le\; i\; \backslash le\; n).$
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 set, finite Mathematical structure, structures. It is closely related to many other ar ...

, a finite set with elements is sometimes called an ''-set'' and a subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...

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, the so-called von Neumann construction, may prefer to use the existence of the bijection $f\; \backslash colon\; S\; \backslash to\; n$, which is equivalent.)
Basic properties

Any proper subset of a finite set ''S'' is finite and has fewer elements than ''S'' itself. As a consequence, there cannot exist abijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...

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, 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#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

) alone.
The axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of axiomatic set theory, set theory that states that every countable collection of empty set, non-empty set (mathematics), sets must have a choice function. Tha ...

, 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
In mathematics, a surjective function (also known as surjection, or onto function) is a Function (mathematics), function that every element can be mapped from element so that . In other words, every element of the function's codomain is the Im ...

(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
:$,\; S\; \backslash cup\; T,\; \backslash le\; ,\; S,\; +\; ,\; T,\; .$
In fact, by the inclusion–exclusion principle
In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union (set theory), union of two finite sets; symbolically exp ...

:
:$,\; S\; \backslash cup\; T,\; =\; ,\; S,\; +\; ,\; T,\; -\; ,\; S\backslash cap\; T,\; .$
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 Set (mathematics), 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 notatio ...

of finite sets is also finite, with:
:$,\; S\; \backslash times\; T,\; =\; ,\; S,\; \backslash times,\; T,\; .$
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
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 ...

''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 (mathematics), set is countable if either it is finite set, 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 fro ...

, 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 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

InZermelo–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 suc ...

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
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Poles, Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.
Biography and studies
Kazimierz Kuratowski was born in Wars ...

) ''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 for the set-theoretical formulation of Kuratowski finiteness.)
# ( Paul Stäckel) ''S'' can be given a total order
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 ...

ing which is well-ordered 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 (mathematics), 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 o ...

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
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

) 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 with respect to inclusion.)
# ''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
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

is also assumed (the axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of axiomatic set theory, set theory that states that every countable collection of empty set, non-empty set (mathematics), sets must have a choice function. Tha ...

is sufficient), then the following conditions are all equivalent:
# ''S'' is a finite set.
# ( Richard Dedekind) 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.
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
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, ...

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 suc ...

with the axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the ...

replaced by its negation.
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. 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 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 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 form ...

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, ar ...

's Type theory
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 mo ...

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 Platonist
Platonism is the philosophy of Plato and school of thought, philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western though ...

s might view particular formal systems as approximating an underlying reality.
Set-theoretic definitions of finiteness

In contexts where the notion ofnatural 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 n ...

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 (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...

to some set of natural numbers of the form $\backslash $. 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.
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
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Poles, Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.
Biography and studies
Kazimierz Kuratowski was born in Wars ...

. (Kuratowski's is the definition used above.)
A set ''S'' is called Dedekind infinite if there exists an injective, non-surjective function $f:S\; \backslash rightarrow\; S$. 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 $x,f(x),f(f(x)),...$. 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 $f(x\_i)\; =\; x\_$ 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 (mathematics), 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 o ...

''P''(''S'') with the structure of a semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilat ...

. 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 theaxiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

, 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 least n ...

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.)
* 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''∣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.
See also

* 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 least n ...

* Peano arithmetic
Notes

References

* * * * * * * * * * * * * * *External links

* {{Set theory Basic concepts in set theory Cardinal numbers