All Topics  
Infinite set

 

 
   Email Print
   Bookmark   Link
 

 

Infinite set
 
 
  Topics Infinite set Topic Home

In set theory
Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
, an infinite set is a set that is not a finite set
Finite set

In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
. Infinite sets may be countable
Countable set

In mathematics, a countable set is a Set with the same cardinality as some subset of the set of natural numbers. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers....
 or uncountable
Uncountable set

In mathematics, an uncountable set is an infinite Set which is too big to be countable set. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the natural numbers....
. Some examples are:

set of natural numbers (whose existence is assured by the axiom of infinity
Axiom of infinity

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....
) is infinite. It is the only set which is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in ZFC
Zermelo–Fraenkel set theory

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 foundations of mathematics....
 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
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 whose cardinality
Cardinality

In mathematics, the cardinality of a set is a measure of the "number of Element of the set". For example, the set A = contains 3 elements, and therefore A has a cardinality of 3....
 is that natural number.

If the axiom of choice
Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinite set many bins and there is no "rule" for which object t...
 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.






Discussion
Ask a question about 'Infinite set'
Start a new discussion about 'Infinite set'
Answer questions from other users
Full Discussion Forum



Encyclopedia


In set theory
Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
, an infinite set is a set that is not a finite set
Finite set

In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
. Infinite sets may be countable
Countable set

In mathematics, a countable set is a Set with the same cardinality as some subset of the set of natural numbers. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers....
 or uncountable
Uncountable set

In mathematics, an uncountable set is an infinite Set which is too big to be countable set. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the natural numbers....
. Some examples are:
  • the set of all integer
    Integer

    The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
    s, , is a countably infinite set; and
  • the set of all real number
    Real number

    In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
    s is an uncountably infinite set
    Uncountable set

    In mathematics, an uncountable set is an infinite Set which is too big to be countable set. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the natural numbers....
    .


Properties

The set of natural numbers (whose existence is assured by the axiom of infinity
Axiom of infinity

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....
) is infinite. It is the only set which is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in ZFC
Zermelo–Fraenkel set theory

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 foundations of mathematics....
 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
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 whose cardinality
Cardinality

In mathematics, the cardinality of a set is a measure of the "number of Element of the set". For example, the set A = contains 3 elements, and therefore A has a cardinality of 3....
 is that natural number.

If the axiom of choice
Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinite set many bins and there is no "rule" for which object t...
 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 powerset 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 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 well-ordered, then it must have a nonempty subset which has no greatest element.

In ZF, a set is infinite if and only if the powerset of its powerset is a Dedekind-infinite set
Dedekind-infinite set

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....
, having a proper subset equinumerous to itself. If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets.

If an infinite set is well-orderable, then it has many well-orderings which are non-isomorphic.

See also

  • Infinity
    Infinity

    Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
  • Aleph number
    Aleph number

    In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets....