set theory Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concern ...

, 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. Infinite sets may be countable or uncountable.

Properties

The set of natural numbers (whose existence is postulated by the axiom of infinity) 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 (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, 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 ...

whose cardinality is that natural number. If the

axiom of choice 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 ...

holds, then a set is infinite if and only if it includes a countable infinite subset. If a

set of sets In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set fa ...

is infinite or contains an infinite element, then its union is infinite. The power set of an infinite set is infinite. Any superset 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'' an infinite set is infinite. The

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...

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 power set 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. Infinite set theory involves proofs and definitions. Important ideas discussed by Burton 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. Candor'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 pi, 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, Kronecker, Cantor, and Bolzano investigated and influenced infinite set theory. Potential historical influences, such as how Prussia's history in the 1800's, resulted in an increase in scholarly mathematical knowledge, including Candor's theory of infinite sets. Mathematicians including Zermelo, Dedekind, Galileo, Kronecker, Cantor, and Bolzano investigated or influenced infinite set theory. Many of these mathematicians either debated infinity or otherwise added to the ideas 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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

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 numbers 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 (e.g. ). The set of all ra ...

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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s is an uncountably infinite set. The set of all irrational numbers is also an uncountably infinite set.

References

External links