In 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,

- Definition and terminology
- Basic properties
- Necessary and sufficient conditions for finiteness
- Foundational issues
- Set-theoretic definitions of finiteness
- Other concepts of finiteness
- See also
- Notes
- References
- External links

is a finite set with five elements. The number of elements of a finite set is a natural number (a non-negative integer) and is called the * cardinality * of the set. A set that is not finite is called *infinite*. For example, the set of all positive integers is infinite:

Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set.

Formally, a set *S* is called **finite** if there exists a bijection

for some natural number *n*. The number *n* is the set's cardinality, denoted as |*S*|. The empty set {} or Ø is considered finite, with cardinality zero.^{ [1] }^{ [2] }^{ [3] }^{ [4] }

If a set is finite, its elements may be written — in many ways — in a sequence:

In combinatorics, a finite set with *n* elements is sometimes called an *n-set* and a subset with *k* elements is called a *k-subset*. For example, the set {5,6,7} is a 3-set – a finite set with three elements – and {6,7} 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 , which is equivalent.)

Any proper subset of a finite set *S* is finite and has fewer elements than *S* itself. As a consequence, there cannot exist a bijection 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) 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

In fact, by the inclusion–exclusion principle:

More generally, the union of any finite number of finite sets is finite. The Cartesian product of finite sets is also finite, with:

Similarly, the Cartesian product of finitely many finite sets is finite. A finite set with *n* elements has 2^{n} distinct subsets. That is, the power set of a finite set is finite, with cardinality 2^{n}.

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, 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.

In Zermelo–Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent:^{[ citation needed ]}

*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 for the set-theoretical formulation of Kuratowski finiteness.) - (Paul Stäckel)
*S*can be given a total ordering 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 of the powerset of*S*is Dedekind-finite (see below).^{ [5] } - 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.^{ [6] }(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 is also assumed (the axiom of countable choice is sufficient^{ [7] }^{[ citation needed ]}), 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.

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 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 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 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 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^{[ citation needed ]} of *set* varying from system to system. Some kinds of Platonists 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 . 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. (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 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*).^{ [8] } 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.

In ZF set theory without the axiom of choice, 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.^{ [9] } (Note that none of these definitions need the set of finite ordinal numbers 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.**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 urelements are found using model theory.^{ [10] }

Most of these finiteness definitions and their names are attributed to Tarski 1954 by Howard & Rubin 1998 , p. 278. However, definitions I, II, III, IV and V were presented in Tarski 1924 , pp. 49, 93, 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.

- ↑ Apostol (1974 , p. 38)
- ↑ Cohn (1981 , p. 7)
- ↑ Labarre (1968 , p. 41)
- ↑ Rudin (1976 , p. 25)
- ↑ The equivalence of the standard numerical definition of finite sets to the Dedekind-finiteness of the power set of the power set was shown in 1912 by Whitehead & Russell 2009 , p. 288. This Whitehead/Russell theorem is described in more modern language by Tarski 1924 , pp. 73–74.
- ↑ Tarski 1924 , pp. 48–58, demonstrated that his definition (which is also known as I-finite) is equivalent to Kuratowski's set-theoretical definition, which he then noted is equivalent to the standard numerical definition via the proof by Kuratowski 1920 , pp. 130–131.
- ↑ Canada, A.; Drabek, P.; Fonda, A. (2005-09-02).
*Handbook of Differential Equations: Ordinary Differential Equations*. Elsevier. ISBN 9780080461083. - ↑ The original paper by Kuratowski 1920 defined a set
*S*to be finite when*P*(*S*)∖{∅} = ⋂{*X*∈*P*(*P*(*S*)∖{∅}); (∀*x*∈*S*, {*x*}∈*X*) and (∀*A*,*B*∈*X*,*A*∪*B*∈*X*)}.

*S*is finite when the set of all non-empty subsets of*S*is equal to the intersection of all classes*X*which satisfy:- all elements of
*X*are non-empty subsets of*S*, - the set {
*x*} is an element of*X*for all*x*in*S*, *X*is closed under pairwise unions.

- ↑ This list of 8 finiteness concepts is presented with this numbering scheme by both Howard & Rubin 1998 , pp. 278–280, and Lévy 1958 , pp. 2–3, although the details of the presentation of the definitions differ in some respects which do not affect the meanings of the concepts.
- ↑ Lévy 1958 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.

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 of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets there exists an indexed family of elements such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

In mathematics, the **axiom of regularity** is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set *A* contains an element that is disjoint from *A*. In first-order logic, the axiom reads:

In the mathematical field of set theory, an **ultrafilter** on a given partially ordered set (poset) *P* is a certain subset of *P,* namely a maximal filter on *P*, that is, a proper filter on *P* that cannot be enlarged to a bigger proper filter on *P*.

In mathematics, an **uncountable set** is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

**Zorn's lemma**, also known as the **Kuratowski–Zorn lemma**, after mathematicians Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain necessarily contains at least one maximal element.

In set theory, the **axiom schema of replacement** is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF.

In set theory, an **infinite set** is a set that is not a finite set. Infinite sets may be countable or uncountable.

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 as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated **ZFC**, where C stands for "choice", and **ZF** refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.

An **enumeration** is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration depend on the discipline of study and the context of a given problem.

In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a **universe** is a collection that contains all the entities one wishes to consider in a given situation.

In elementary set theory, **Cantor's theorem** is a fundamental result which states that, for any set , the set of all subsets of has a strictly greater cardinality than itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with members has a total of subsets, so that if then , and the theorem holds because for all non-negative integers.

In mathematics, two sets or classes *A* and *B* are **equinumerous** if there exists a one-to-one correspondence between them, that is, if there exists a function from *A* to *B* such that for every element *y* of *B*, there is exactly one element *x* of *A* with *f*(*x*) = *y*. Equinumerous sets are said to have the same cardinality. The study of cardinality is often called **equinumerosity** (*equalness-of-number*). The terms **equipollence** (*equalness-of-strength*) and **equipotence** (*equalness-of-power*) are sometimes used instead.

The **axiom of countable choice** or **axiom of denumerable choice**, denoted **AC _{ω}**, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function

In mathematics, a **non-measurable set** is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In ZF, choice entails that non-measurable subsets of exists.

In mathematics, a set *A* is **Dedekind-infinite** if some proper subset *B* of *A* is equinumerous to *A*. Explicitly, this means that there exists a bijective function from *A* onto some proper subset *B* of *A*. A set is **Dedekind-finite** if it is not Dedekind-infinite. Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.

Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.

This article contains a discussion of **paradoxes of set theory**. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory.

In set theory, an **ordinal number**, or **ordinal**, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

In mathematics a group is a set together with a binary operation on the set called **multiplication** that obeys the group axioms. The axiom of choice is an axiom of ZFC set theory which in one form states that every set can be wellordered.

This is a **glossary of set theory**.

- Apostol, Tom M. (1974),
*Mathematical Analysis*(2nd ed.), Menlo Park: Addison-Wesley, LCCN 72011473 - Cohn, Paul Moritz, F.R.S. (1981),
*Universal Algebra*, Dordrecht: D. Reidel, ISBN 90-277-1254-9, LCCN 80-29568 - Dedekind, Richard (2012),
*Was sind und was sollen die Zahlen?*, Cambridge Library Collection (Paperback ed.), Cambridge, UK: Cambridge University Press, ISBN 978-1-108-05038-8 - Dedekind, Richard (1963),
*Essays on the Theory of Numbers*, Dover Books on Mathematics, Beman, Wooster Woodruff (Paperback ed.), Dover Publications Inc., ISBN 0-486-21010-3 - Herrlich, Horst (2006),
*Axiom of Choice*, Lecture Notes in Math. 1876, Berlin: Springer-Verlag, ISBN 3-540-30989-6 - Howard, Paul; Rubin, Jean E. (1998).
*Consequences of the axiom of choice*. Providence, Rhode Island: American Mathematical Society. ISBN 9780821809778. - Kuratowski, Kazimierz (1920), "Sur la notion d'ensemble fini" (PDF),
*Fundamenta Mathematicae*,**1**: 129–131, doi:10.4064/fm-1-1-129-131 - Labarre, Jr., Anthony E. (1968),
*Intermediate Mathematical Analysis*, New York: Holt, Rinehart and Winston, LCCN 68019130 - Lévy, Azriel (1958). "The independence of various definitions of finiteness" (PDF).
*Fundamenta Mathematicae*.**46**: 1–13. doi:10.4064/fm-46-1-1-13. - Rudin, Walter (1976),
*Principles Of Mathematical Analysis*(3rd ed.), New York: McGraw-Hill, ISBN 0-07-054235-X - Suppes, Patrick (1972) [1960],
*Axiomatic Set Theory*, Dover Books on Mathematics (Paperback ed.), Dover Publications Inc., ISBN 0-486-61630-4 - Tarski, Alfred (1924). "Sur les ensembles finis" (PDF).
*Fundamenta Mathematicae*.**6**: 45–95. doi:10.4064/fm-6-1-45-95. - Tarski, Alfred (1954). "Theorems on the existence of successors of cardinals, and the axiom of choice".
*Nederl. Akad. Wetensch. Proc. Ser. A., Indagationes Math*.**16**: 26–32. doi:10.1016/S1385-7258(54)50005-3. MR 0060555. - Whitehead, Alfred North; Russell, Bertrand (February 2009) [1912].
*Principia Mathematica*. Volume Two. Merchant Books. ISBN 978-1-60386-183-0.

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