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That is, it is something which "goes on without end". Infinity is denoted by this symbol:<math>\infty\,</math>, looking like an 8 on its side.
That is, it is something which "goes on without end". Infinity is denoted by this symbol:<math>\infty\,</math>, looking like an 8 on its side.
Infinity comes up in many aspects of mathematical discourse and is often treated as a number, but it is not a [[real number|real number]]. In mathematical contexts, infinity often appears as a limit, as an integral, as the measure (size) of a set in Euclidean space, and as the cardinality (size) of sets in general.
Infinity comes up in many aspects of mathematical discourse and is often treated as a number, but it is not a [[real number]]. In mathematical contexts, infinity often appears as a limit, as an integral, as the measure (size) of a set in Euclidean space, and as the cardinality (size) of sets in general.
The simplest place to see this seeming paradox is in the fact that all integers are finite, but that there are an infinite number of them.
The simplest place to see this seeming paradox is in the fact that all integers are finite, but that there are an infinite number of them.
== Classification of infinities ==
== Classification of infinities ==
In modern mathematics, it is recognized that there is no single concept of 'infinity' - there are many different infinities. We can divide these into some broad classes. Firstly, there are the cardinals, which are used to measure the sizes of sets; and then there are ordinals, which are used to measure the position of an item in an ordered list. For finite quantities, we can use the same numbers for both - a race can have 3 participants, and you can come 3<sup>rd</sup> in the race. But, with infinite quantities, this no longer applies, the same numbers can now be used for both purposes. Thus, the smallest infinite cardinal is aleph-null, but the smallest infinite ordinal is omega.
In modern mathematics, it is recognized that there is no single concept of 'infinity' - there are many different infinities. We can divide these into some broad classes. Firstly, there are the cardinals, which are used to measure the sizes of sets; and then there are ordinals, which are used to measure the position of an item in an ordered list. For finite quantities, we can use the same numbers for both - a race can have 3 participants, and you can come 3rd in the race. But, with infinite quantities, this no longer applies, the same numbers can now be used for both purposes. Thus, the smallest infinite cardinal is aleph-null, but the smallest infinite ordinal is omega.
=== Cardinals ===
=== Cardinals ===
==Ordinals==
==Ordinals==
Numbers serve two distinct purposes - to measure the size of sets, and to measure the position of an item in an ordering. Thus, we may speak of a race having 2, 3, 4, 5, etc. contestants, and we can speak of the contestants as having come 1<sup>st</sup>, 2<sup>nd</sup>, 3<sup>rd</sup> etc. (or even 0<sup>th</sup>, if one is a mathematician!) When we measure the number of elements in a set, we are using '''cardinal''' numbers; when we measure an item's position in an ordering, we are using '''ordinal''' numbers. For finite quantities, it does not make that much difference, since for finite quantities we can use the same numbers to serve both purposes. But for transfinite quantities, that is no longer the case - we can no longer use the same numbers as both cardinals and ordinals. Thus, the smallest transfinite cardinal
Numbers serve two distinct purposes - to measure the size of sets, and to measure the position of an item in an ordering. Thus, we may speak of a race having 2, 3, 4, 5, etc. contestants, and we can speak of the contestants as having come 1st, 2nd, 3rd etc. (or even 0th, if one is a mathematician!) When we measure the number of elements in a set, we are using '''cardinal''' numbers; when we measure an item's position in an ordering, we are using '''ordinal''' numbers. For finite quantities, it does not make that much difference, since for finite quantities we can use the same numbers to serve both purposes. But for transfinite quantities, that is no longer the case - we can no longer use the same numbers as both cardinals and ordinals. Thus, the smallest transfinite cardinal
is <math>\aleph_{0}</math>, but the smallest transfinite ordinal is <math>\omega</math>.
is <math>\aleph_{0}</math>, but the smallest transfinite ordinal is <math>\omega</math>.