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{{Math-h}}

{{Math-h}}

A dictionary definition of '''Infinity''' is an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount"<ref> [http://www.merriam-webster.com/dictionary/infinity Merriam-Webster dictionary]</ref>

'''Infinity''' may be defined as an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount."<ref>[http://www.merriam-webster.com/dictionary/infinity Merriam-Webster dictionary]</ref>

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|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.

*Why are all integers finite?  Because, if infinity were an integer, what would <math>\infty+1\,</math> be?

*Why are all integers finite?  Because, if infinity were an integer, what would <math>\infty+1\,</math> be?

*Why is the ''set'' of all integers an infinite set?  Because they go on forever. If the set were finite, there would be a biggest integer. But that can't be, because we can always add one to any integer. This kind of thinking is often schoolchildren's first introduction to [[logical reasoning]].   

*Why is the ''set'' of all integers an infinite set?  Because they go on forever. If the set were finite, there would be a biggest integer. But that can't be, because we can always add one to any integer. This kind of thinking is often schoolchildren's first introduction to [[logical reasoning]].   

While infinity is not a number, it appears in many other contexts.  As we have seen, the size of the set of integers is infinite. We say that the [[cardinality]] of the integers (this set is often denoted <math>\mathbb{Z}</math>) is infinite.  The rational numbers (<math>\mathbb{Q}</math>) and the real numbers (<math>\mathbb{R}</math>) also have infinite cardinality.  An interesting result from set theory (see below) is that <math>\mathbb{Z}</math> and <math>\mathbb{Q}</math> have the same cardinality, while <math>\mathbb{R}</math> has larger cardinality than the other two.

While infinity is not a number, it appears in many other contexts.  As we have seen, the size of the set of integers is infinite. We say that the [[cardinality]] of the integers (this set is often denoted <math>\mathbb{Z}</math>) is infinite.  The rational numbers (<math>\mathbb{Q}</math>) and the real numbers (<math>\mathbb{R}</math>) also have infinite cardinality.  An interesting result from set theory (see below) is that <math>\mathbb{Z}</math> and <math>\mathbb{Q}</math> have the same cardinality, while <math>\mathbb{R}</math> has larger cardinality than the other two.

Another place where infinity arises is in limits. We can say that "the limit of <math>1/x\,</math> as <math>x\,</math> approaches zero is infinity", or "the limit of <math>e^{-x}\,</math> as <math>x\,</math> approaches infinity is zero", but this is because infinity has a special meaning in the context of limits.  See [[limit (mathematics)|limit]] for discussion of this.

Another place where infinity arises is in limits. We can say that "the limit of <math>1/x\,</math> as <math>x\,</math> approaches zero is infinity", or "the limit of <math>e^{-x}\,</math> as <math>x\,</math> approaches infinity is zero", but this is because infinity has a special meaning in the context of limits.  See [[limit (mathematics)|limit]] for discussion of this.

We can also say that "the measure (size) of the set of reals is infinity", or that certain integrals are infinite, but this is because of special properties of measures and integrals.

We can also say that "the measure (size) of the set of reals is infinity", or that certain integrals are infinite, but this is because of special properties of measures and integrals.

So infinity might arise in statements like these:

So infinity might arise in statements like these:

*<math>\frac{1}{0} = \infty\ \ </math>NO!  This isn't allowed!  Infinity is not a number, and division by zero is illegal!

*<math>\frac{1}{0} = \infty\ \ </math>NO!  This isn't allowed!  Infinity is not a number, and division by zero is illegal!

*<math>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math>Yes, sort of. One could say that "the limit is infinite", since that includes both positive and negative infinite values.  But to say that the limit is "equal to infinity", one would have to say that this is the limit from the right. The limit from the left is "minus infinity", that is, unboundedly negative. This sort of statement, in terms of limits, is what was presumably meant by the incorrect statement above.

*<math>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math>Yes, sort of. One could say that "the limit is infinite", since that includes both positive and negative infinite values.  But to say that the limit is "equal to infinity", one would have to say that this is the limit from the right. The limit from the left is "minus infinity", that is, unboundedly negative. This sort of statement, in terms of limits, is what was presumably meant by the incorrect statement above.

*<math>\int_0^1\frac{1}{x} = \infty\ \ </math>Infinity has a special meaning for integrals.

*<math>\int_0^1\frac{1}{x} = \infty\ \ </math>Infinity has a special meaning for integrals.

*<math>\|\mathbb{Z}\| = \infty\ \ </math>The cardinality of the integers is infinite.

*<math>\|\mathbb{Z}\| = \infty\ \ </math>The cardinality of the integers is infinite.

In measure theory, one sometimes defines the "extended reals", allowing plus and minus infinity to be considered to be numbers, but this is a special construction, which makes certain arithmetical operations impossible.  It can't be done in general.

In measure theory, one sometimes defines the "extended reals", allowing plus and minus infinity to be considered to be numbers, but this is a special construction, which makes certain arithmetical operations impossible.  It can't be done in general.

Another place where infinity is considered acceptable is in [[Floating-point]] arithmetic in computers.  The IEEE-754 standard (which all modern computers support) allows the values <math>\infty\,</math> and <math>- \infty\,</math>. But their treatment is very different from other numbers, and computer floating-point numbers don't faithfully model the real numbers in any case.

Another place where infinity is considered acceptable is in [[Floating-point]] arithmetic in computers.  The IEEE-754 standard (which all modern computers support) allows the values <math>\infty\,</math> and <math>- \infty\,</math>. But their treatment is very different from other numbers, and computer floating-point numbers don't faithfully model the real numbers in any case.

Also, in some non-standard models of [[Peano Arithmetic]], <math>\infty\,</math> is treated as an actual number. But, once again, this is not standard mathematics.

Also, in some non-standard models of [[Peano Arithmetic]], <math>\infty\,</math> is treated as an actual number. But, once again, this is not standard mathematics.

[[Georg Cantor]]'s [[diagonalization|diagonal argument]] is an elegant proof demonstrating that the (infinite) cardinality of [[real number]]s is greater than the (infinite) cardinality of [[countable]] [[integer]]s.  The essence of the argument is that in any proposed list of all real numbers, a new real number not in the list can be constructed by taking the digits in a [[diagonal]] through the list and changing them to construct a new real number that differs from the nth entry at the nth position right of the decimal point.   

[[Georg Cantor]]'s [[diagonalization|diagonal argument]] is an elegant proof demonstrating that the (infinite) cardinality of [[real number]]s is greater than the (infinite) cardinality of [[countable]] [[integer]]s.  The essence of the argument is that in any proposed list of all real numbers, a new real number not in the list can be constructed by taking the digits in a [[diagonal]] through the list and changing them to construct a new real number that differs from the nth entry at the nth position right of the decimal point.   

To formalize the concepts of countably and uncountably infinite sets, we need the [[set theory]] concept of [[cardinality]]. Using this concept it can be shown that there are infinitely many distinct infinite cardinalities.

To formalize the concepts of countably and uncountably infinite sets, we need the [[set theory]] concept of [[cardinality]]. Using this concept it can be shown that there are infinitely many distinct infinite cardinalities.

In [[Zermelo-Fraenkel]] set theory, there is the [[Axiom of Infinity]], asserting the existence of an infinite set. The set that it creates is essentially the same as the integers.  Some "constructive" mathematicians work without this axiom, and determine which results may be proved without assuming it.  

In [[Zermelo-Fraenkel]] set theory, there is the [[Axiom of Infinity]], asserting the existence of an infinite set. The set that it creates is essentially the same as the integers.  Some "constructive" mathematicians work without this axiom, and determine which results may be proved without assuming it.  

== Classification of infinities ==

== Classification of infinities ==