Definition:Infinity

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Definition

Informally, the term infinity is used to mean some infinite number, but this concept falls very far short of a usable definition.

The symbol $\infty$ (supposedly invented by John Wallis) is often used in this context to mean an infinite number.

However, outside of its formal use in the definition of limits its use is strongly discouraged until you know what you're talking about.


It is defined as having the following properties:

\(\ds \forall n \in \Z: \, \) \(\ds n\) \(<\) \(\ds \infty\)
\(\ds \forall n \in \Z: \, \) \(\ds n + \infty\) \(=\) \(\ds \infty\)
\(\ds \forall n \in \Z: \, \) \(\ds n \times \infty\) \(=\) \(\ds \infty\)
\(\ds \infty^2\) \(=\) \(\ds \infty\)


Similarly, the quantity written as $-\infty$ is defined as having the following properties:

\(\ds \forall n \in \Z: \, \) \(\ds -\infty\) \(<\) \(\ds n\)
\(\ds \forall n \in \Z: \, \) \(\ds -\infty + n\) \(=\) \(\ds -\infty\)
\(\ds \forall n \in \Z: \, \) \(\ds -\infty \times n\) \(=\) \(\ds -\infty\)
\(\ds \paren {-\infty}^2\) \(=\) \(\ds -\infty\)


The latter result seems wrong when you think of the rule that a negative number squared equals a positive number, but remember that infinity is not exactly a number as such.


Actual Infinity

An actual infinity is a given object that consists of an infinite number of elements.


Potential Infinity

A potential infinity is a concept of infinity in which an endless process produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps.


Ad Infinitum

The term ad infinitum means:

endlessly
repeating indefinitely
generating an infinite sequence of terms

and so on.

Sometimes it has the suggestion of an endless loop.


Also, ad infinitum can be used when an infinite sequence is defined merely by showing some initial terms but without explicitly specifying either with a formula or recursively.


Examples

Limit of Reciprocal

The real function $f: \R \to \R$ defined as:

$\forall x \in \R: \map f x = \dfrac 1 x$

becomes larger in absolute value as $x$ approaches $0$.


Thus:

$\ds \lim_{x \mathop \to 0^+} \map f x \to +\infty$

That is, for every $C \in \R_{>0}$ there exists $a \in \R_{>0}$ such that $\map f x > C$ when $0 < x < a$.

When $\map f x \to +\infty$, $\map f x$ is said to be positively infinite.


Similarly:

$\ds \lim_{x \mathop \to 0^-} \map f x \to -\infty$

That is, for every $C \in \R_{>0}$ there exists $a \in \R_{>0}$ such that $\map f x > -C$ when $-a < x < 0$.

When $\map f x \to -\infty$, $\map f x$ is said to be negatively infinite.


Also see

  • Results about infinity can be found here.


Historical Note

The concept of infinity dates back to Zeno of Elea in the $5$th century BCE and Eudoxus of Cnidus in the $4$th century BCE.

It has bothered scientists, mathematicians and philosophers since the time of Plato (who accepted the concept as realisable) Aristotle (who did not).

The symbol $\infty$ for infinity was introduced by John Wallis in the $17$th century.

It was Georg Cantor in the $1870$s who finally made the bold step of positing the actual existence of infinite sets as mathematical objects which paved the way towards a proper understanding of infinity.


Sources

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