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In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, . Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property . . , n} where n is a natural number. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an (The value n = 0 is allowed; that is, the empty set is finite. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members ) An infinite set is a set which is not finite. In Set theory, an infinite set is a set that is not a Finite set.

Equivalently, a set is finite if its cardinality, i. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" e. , the number of its elements, is a natural number. More specifically, a set whose cardinality is n is also called an n-set. In Mathematics, an n-set is a set containing exactly n elements where n is a Natural number. 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. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In contrast, the set of all prime numbers has cardinality ℵ0, so it is infinite. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

A set is called Dedekind-finite if there exists no bijection between the set and any of its proper subsets. In Mathematics, a set A is Dedekind-infinite if some proper Subset B of A is Equinumerous to A. 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. In Mathematics, the axiom of dependent choices, denoted DC, is a weak form of the Axiom of choice (AC which is still sufficient to develop most of In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. ↔ Otherwise, paradoxically, there may be infinite Dedekind-finite sets (see Foundational issues below). In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2.

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. )

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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. In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom of infinity is one of the Axioms

Necessary and sufficient conditions for finiteness

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

  1. S is a finite set. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common 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. Kazimierz Kuratowski ( Warsaw, February 2, 1896 &ndash June 18, 1980) was a Polish Mathematician and Logician (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. Paul Stäckel ( 20 August 1862 — 12 December 1919) was a German Mathematician, active in the areas of Differential In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every 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). In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S)
  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. Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California
  7. 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 In other words, the well-orderings on S have exactly one order type. In Mathematics, especially in Set theory, two Ordered sets XY are said to have the same order type just when they are Order isomorphic

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

  1. S is a finite set. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.
  2. (Richard Dedekind) Every function from S one-to-one into itself is onto. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important
  3. Every function from S onto itself is one-to-one.
  4. Every partial ordering of S contains a maximal element. In Mathematics, especially in Order theory, a maximal element of a subset S of some Partially ordered set is an element of S that

Foundational issues

Georg Cantor initiated his theory of sets in order to provide a mathematical treatment of infinite sets. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. 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. In the Philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed 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. In Mathematics, hereditarily finite sets are defined recursively as Finite sets containing only hereditarily finite sets (with the Empty set as Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom of infinity is one of the Axioms

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. In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most 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 Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural In particular, there exists a plethora of so-called non-standard models of both theories. See also Formal interpretation In Model theory, a discipline within Mathematical logic, a non-standard model is a 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. In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist 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. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In the Foundations of mathematics, Von Neumann–Bernays–Gödel set theory ( NBG) is an Axiomatic set theory that is a Conservative extension Non-well founded set theories are variants of Axiomatic set theory which allow sets to contain themselves and otherwise violate the rule of Well-foundedness. Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian In Mathematics, Logic and Computer science, type theory is any of several Formal systems that can serve as alternatives to Naive set theory One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic. Intuitionistic logic, or constructivist logic, is the Symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer

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. Platonism is the Philosophy of Plato or the name of other philosophical systems considered closely derived from it

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 {x | x < n}. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every Such an approach requires a structural definition of finiteness that does not depend on natural numbers. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

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). Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Kazimierz Kuratowski ( Warsaw, February 2, 1896 &ndash June 18, 1980) was a Polish Mathematician and Logician

Call a set S Dedekind infinite if there exists an injective, non-surjective function f:S \rightarrow S. In Mathematics, a set A is Dedekind-infinite if some proper Subset B of A is Equinumerous to A. 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(x),f(f(x)),. In Mathematics, a set A is Dedekind-infinite if some proper Subset B of A is Equinumerous to A. . . . Conversely, given a sequence in S consisting of elements x1,x2,x3,. . . , we can define a function f such that on elements in the sequence f(xi) = xi + 1 and f behaves like the identity function otherwise. Thus Dedekind infinite sets contain subsets that correspond bijectively with the natural numbers. In Mathematics, a set A is Dedekind-infinite if some proper Subset B of A is Equinumerous to A. 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. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) A semilattice is a mathematical concept with two definitions one as a type of Ordered set, the other as an Algebraic structure. 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. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Mathematics, a singleton is a set with exactly one element

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:

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.

See also

References

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