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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]].<ref>Merrian-Webster defines infinity as an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount." [http://www.merriam-webster.com/dictionary/infinity Merriam-Webster dictionary]  Put another way, infinity is something which "goes on without end."</ref>

*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.<ref>Merrian-Webster defines infinity as an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount." [http://www.merriam-webster.com/dictionary/infinity Merriam-Webster dictionary]  Put another way, infinity is something which "goes on without end."</ref>

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.

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 ==