Infinity

<math>\frac{d}{dx} \sin x=?\,</math>

This article/section deals with mathematical concepts appropriate for late high school or early college.

A dictionary definition of Infinity is an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount"

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. But not as a number.

INFINITY IS NOT A NUMBER!

But it comes up in many other mathematical contexts—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.

• 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. There can't be a "biggest" integer, 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.

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 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.

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>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math>Yes. This 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>|\mathbb{Z}| = \infty\ \ </math>The cardinality of the integers is infinite.
• <math>\mu(\mathbb{R}) = \infty\ \ </math>The (Lebesgue) measure of the reals 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.

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 diagonal argument is an elegant proof demonstrating that the (infinite) cardinality of real numbers is greater than the (infinite) cardinality of countable integers. 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.

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.