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# Finite set

In mathematics, a finite set is a set that has a finite number of elements. 

63 relations: Addison-Wesley, Alfred Tarski, Axiom of choice, Axiom of countable choice, Axiom of infinity, Bertrand Russell, Bijection, Cardinality, Cartesian product, Combinatorics, Countable set, Counting, D. Reidel, Dedekind-infinite set, Element (mathematics), Empty set, Finitism, FinSet, Formal system, Formalism (philosophy of mathematics), Free lattice, Fundamenta Mathematicae, Gödel's incompleteness theorems, Georg Cantor, Hereditarily finite set, Holt McDougal, Infinite set, Injective function, Integer, Intuitionistic logic, Join and meet, Kazimierz Kuratowski, Mathematics, Maximal and minimal elements, McGraw-Hill Education, Model theory, Natural number, Non-standard model, Non-well-founded set theory, Order isomorphism, Order type, Ordinal number, Partially ordered set, Paul Stäckel, Peano axioms, Philosophy of mathematics, Pigeonhole principle, Power set, Richard Dedekind, Semilattice, ... Expand index (13 more) »

Addison-Wesley is a publisher of textbooks and computer literature.

## Alfred Tarski

Alfred Tarski (January 14, 1901 &ndash; October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews,, School of Mathematics and Statistics, University of St Andrews.

## Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

## Axiom of countable choice

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

## 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.

## Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 &ndash; 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic, political activist, and Nobel laureate.

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

## Cartesian product

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.

## 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.

## Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

## Counting

Counting is the action of finding the number of elements of a finite set of objects.

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## Dedekind-infinite set

In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) 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.

## Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

## Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Finitism

Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects.

## FinSet

In the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them.

## Formal system

A formal system is the name of a logic system usually defined in the mathematical way.

## Formalism (philosophy of mathematics)

In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be considered to be statements about the consequences of certain string manipulation rules.

## Free lattice

In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice.

## Fundamenta Mathematicae

Fundamenta Mathematicae is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems.

## Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic.

## Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.

## Hereditarily finite set

In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets.

## Holt McDougal

Holt McDougal is an American publishing company, a division of Houghton Mifflin Harcourt, that specializes in textbooks for use in secondary schools.

## Infinite set

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

## Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Intuitionistic logic

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof.

## Join and meet

In a partially ordered set P, the join and meet of a subset S are respectively the supremum (least upper bound) of S, denoted &#8897;S, and infimum (greatest lower bound) of S, denoted &#8896;S.

## Kazimierz Kuratowski

Kazimierz Kuratowski (Polish pronunciation:, 2 February 1896 – 18 June 1980) was a Polish mathematician and logician.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Maximal and minimal elements

In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.

## McGraw-Hill Education

McGraw-Hill Education (MHE) is a learning science company and one of the "big three" educational publishers that provides customized educational content, software, and services for pre-K through postgraduate education.

## Model theory

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

## Natural number

In mathematics, the natural numbers are those 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").

## Non-standard model

In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model).

## Non-well-founded set theory

Non-well-founded set theories are variants of axiomatic set theory that allow sets to contain themselves and otherwise violate the rule of well-foundedness.

## Order isomorphism

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).

## Order type

In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when they are order isomorphic, that is, when there exists a bijection (each element matches exactly one in the other set) f: X &rarr; Y such that both f and its inverse are strictly increasing (order preserving i.e. the matching elements are also in the correct order).

## Ordinal number

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.

## Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

## 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.

## Peano axioms

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.

## Philosophy of mathematics

The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics, and purports to provide a viewpoint of the nature and methodology of mathematics, and to understand the place of mathematics in people's lives.

## 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.

## Power set

In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.

## Richard Dedekind

Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.

## Semilattice

In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

## Sign (mathematics)

In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.

## Singleton (mathematics)

In mathematics, a singleton, also known as a unit set, is a set with exactly one element.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

## Type theory

In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics.

## Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

## Von Neumann–Bernays–Gödel set theory

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel set theory (ZFC).

## 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.

## Zermelo–Fraenkel set theory

In mathematics, 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.

## References

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