set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

, an infinite set is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

that is not a

finite set 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, is a finite set with five elements. Th ...

. Infinite sets may be

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 ...

uncountable In mathematics, an uncountable set, informally, 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 tha ...

Properties

The set of

natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

(whose existence is postulated by 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 ...

) is infinite. It is the only set that is directly required by the

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s to be infinite. The existence of any other infinite set can be proved in

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 ...

(ZFC), but only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natural number, the set has a

subset In mathematics, a 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 a ...

whose

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

is that natural number. If the

axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

holds, then a set is infinite if and only if it includes a countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite. The

power set In mathematics, the power set (or powerset) of a 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 of any set is po ...

of an infinite set is infinite. Any

superset In mathematics, a set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset ...

of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped ''

onto In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

'' an infinite set is infinite. The

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets, each containing at least two elements, is either empty or infinite; if the axiom of choice holds, then it is infinite. If an infinite set is a well-ordered set, then it must have a nonempty, nontrivial subset that has no greatest element. In ZF, a set is infinite if and only if the

of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets. If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.

History

Important ideas discussed by David Burton in his book ''The History of Mathematics: An Introduction'' include how to define "elements" or parts of a set, how to define unique elements in the set, and how to prove infinity. Burton also discusses proofs for different types of infinity, including countable and uncountable sets. Topics used when comparing infinite and finite sets include ordered sets, cardinality, equivalency, coordinate planes, universal sets, mapping, subsets, continuity, and transcendence. Cantor's set ideas were influenced by trigonometry and irrational numbers. Other key ideas in infinite set theory mentioned by Burton, Paula, Narli and Rodger include real numbers such as , integers, and Euler's number. Both Burton and Rogers use finite sets to start to explain infinite sets using proof concepts such as mapping, proof by induction, or proof by contradiction. Mathematical trees can also be used to understand infinite sets. Burton also discusses proofs of infinite sets including ideas such as unions and subsets. In Chapter 12 of ''The History of Mathematics: An Introduction'', Burton emphasizes how mathematicians such as Zermelo, Dedekind,

Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...

, Kronecker, Cantor, and

Bolzano Bolzano ( ; ; or ) is the capital city of South Tyrol (officially the province of Bolzano), Northern Italy. With a population of 108,245, Bolzano is also by far the largest city in South Tyrol and the third largest in historical Tyrol. The ...

investigated and influenced infinite set theory. Many of these mathematicians either debated infinity or otherwise added to the ideas of infinite sets. Potential historical influences, such as how Prussia's history in the 1800s, resulted in an increase in scholarly mathematical knowledge, including Cantor's theory of infinite sets. One potential application of infinite set theory is in genetics and biology.

Examples

Countably infinite sets

The set of all

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s, is a countably infinite set. The set of all even integers is also a countably infinite set, even if it is a proper subset of the integers. The set of all

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s is a countably infinite set as there is a bijection to the set of integers.

Uncountably infinite sets

The set of all

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s is an uncountably infinite set. The set of all

irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...

s is also an uncountably infinite set. The set of all subsets of the integers is uncountably infinite.

References

External links

A Crash Course in the Mathematics Of Infinite Sets
{{Mathematical logic Cardinal numbers