In

A Crash Course in the Mathematics Of Infinite Sets

{{Mathematical logic Cardinal numbers

set theory
Set theory is the branch of mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and ...

, an infinite set is a set that is not a finite set. Infinite sets may be countable
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

or uncountable.
Properties

The set ofnatural numbers
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

(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
Ancient Greek includes the forms of the ...

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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

whose cardinality is that natural number.
If the axiom of choice 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 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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

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
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

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 allinteger
An integer is the number zero (), a positive natural number
In mathematics, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples are the natural number ...

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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

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

The set of allreal number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

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

* Aleph number *Cardinal number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

*Ordinal number
In set theory
Set theory is the branch of mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related str ...

References

External links

A Crash Course in the Mathematics Of Infinite Sets

{{Mathematical logic Cardinal numbers