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Encyclopedia > 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.) All finite sets are countable [1], but not all countable sets are finite. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A bijective function. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics the term countable set is used to describe the size of a set, e. ...


Equivalently, a set is finite if its cardinality, i.e. the number of its elements, is a natural number. For instance, the set of integers between -15 and 3 (excluding the end points) is finite, since it has 17 elements. The set of all prime numbers is not finite. Infinite sets are sets which are not finite. In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... The integers are commonly denoted by the above symbol. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ... In set theory, an infinite set is a set that is not a finite set. ...


A set is called Dedekind finite if there exists no bijection between the set and any of its proper subsets. Failure of the axiom of choice can give rise to infinite, Dedekind finite sets, as discussed in the section on foundational issues. In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...


In the literature, countably infinite means that a set admits a bijection with the set of natural numbers. The adjective countable may or may not include finite sets, depending on the whim of the author.

Contents

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 of Zermelo-Fraenkel set theory. ...


Necessary and sufficient conditions for finiteness

In ZF, the following conditions are all equivalent: Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...

  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: Kazimierz Kuratowski (born February 2, 1896, Warsaw, died June 18, 1980, Warsaw) was a Polish mathematician. ... Paul Stäckel (20 August 1862 — 12 December 1919) was a German mathematician, active in the areas of differential geometry, number theory, and non-Euclidean geometry. ... In mathematics, a well-order (or well-ordering) 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. ... In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ... Alfred Tarski (January 14, 1901, Warsaw Poland – October 26, 1983, Berkeley California) was a logician and mathematician of considerable philosophical importance. ... 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. ... In mathematics, especially in set theory, ordinals may be used to label the elements of any given well-ordered set (the smallest element being labeled 0, the one after that 1, the next one 2, and so on) and to measure the length of the whole set by the least... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

  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.

Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. ...

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. Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... 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 of Zermelo-Fraenkel set theory. ...


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 Godel'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-stardard models of both theories. A seeming paradox, non-stardard 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. In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ...


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. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ... In foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiom system for set theory designed to yield the same results as Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC), but with only a finite number of axioms, that is without axiom schemas. ... 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, logician, and mathematician. ... At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into collections called types. ...


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. The term formalist can have many applications: The Chambers 1994 edition Dictionary indicates a pejorative quality, a person having an exaggerated regard to rules or established usages. In the philosophy of mathematics a formalist is a person who belongs to the school of formalism, a certain mathematical-philosophical doctrine which... Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ...


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 of the form {x | x < n}. 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. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A bijective function. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In mathematics, a well-order (or well-ordering) 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. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...


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). Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... Kazimierz Kuratowski (born February 2, 1896, Warsaw, died June 18, 1980, Warsaw) was a Polish mathematician. ...


Dedekind treats infinitude as the positive notion and finiteness as its negation. 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(x),f(f(x)),.... 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) = f(xi + 1) 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. In set theory a set S is Dedekind-infinite if there is a bijective function from S to some proper subset of S, or equivalently if there is an injective function from the natural numbers into S. In the absence of choice, Dedekind-infinite is a stronger condition than merely... In set theory a set S is Dedekind-infinite if there is a bijective function from S to some proper subset of S, or equivalently if there is an injective function from the natural numbers into S. In the absence of choice, Dedekind-infinite is a stronger condition than merely... In set theory a set S is Dedekind-infinite if there is a bijective function from S to some proper subset of S, or equivalently if there is an injective function from the natural numbers into S. In the absence of choice, Dedekind-infinite is a stronger condition than merely...


Kuratowski treats finiteness as the positive notion. 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. In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ... A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... 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:

  • 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, Kurotowski 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 choice 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 Kurotowski finiteness would fail for the same set of socks.


Footnotes

  1. ^ Some authors use "countable" to mean "countably infinite", and thus do not consider finite sets to be countable.

In set theory, an infinite set is a set that is not a finite set. ...

See also

The infinity symbol ∞ in several typefaces. ... In mathematics, hereditarily finite sets are defined recursively as finite sets containing only hereditarily finite sets (with the empty set as a base case). ...

References

  • Patrick Suppes, Axiomatic Set Theory, D. Van Nostrand Company, Inc., 1960

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