What exactly is infinity?

On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and $\infty - \infty$.

From what I know, given $x$ being any number, excluding $0$, $\frac{x}{x} = 1$ is true.

So just what, exactly, is $\infty$?

Just to be clear: Infinity is not a number.

Also, it is likely that there is no "exact" (agreed upon) characterization of infinity.
In a sense, casually put: $$\infty = \{\text{that which is NOT finite}\}$$

Definition: Infinity refers to something without any limit, and is a concept relevant in a number of fields, predominantly mathematics and physics. The English word infinity derives from Latin infinitas, which can be translated as "unboundedness", itself derived from the Greek word apeiros, meaning "endless".

"In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities). For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite."

-Wikipedia: Infinity

For a more expansive discussion on infinity, y

• You might want to read: "Counting to Infinity".
  In short, it seems that there are "different" infinities (perhaps infinitely many infinities!), some larger than others.
• One enchanting video may be of interest: See YouTube on Infinity.
• "Hear" BBC Radio on Infinity.

EDIT: You might want to be assured that you are not the only one grappling with the concept of infinity: Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of the philosophical and mathematical schools of constructivism and intuitionism.

I personally had found infinity to be a bit confusing, until I had a professor that always associated it with the phrase "arbitrary large". I think that "arbitrary large" (or "unbounded") is a good way to conceptualize infinity. What does it mean for there to be an infinite number of primes? It means that you can find arbitrarily large prime numbers. What does $$\lim_{n\to\infty}\frac{1}{n} = 0$$ mean? It means that as $n$ gets arbitrarily large, $\frac{1}{n}$ gets arbitrarily close to 0.

Mathematically, this can be expressed by the following statement: A set $A$ is infinite if and only if, given any finite subset $B\subseteq A$, there is always an element $x\in A$ such that $x\not\in B$. In other words, no matter how big of a finite subset you choose, there is always a bigger one (in this case $B\cup\{x\}$).

The positive real numbers make a nice abstraction of the notion of length.

The infinity on the real line represents an abstract notion of "being longer than any other length". You can think of it formally as being larger than any finite length: something has "infinite" length if it is longer than an object of length $1$, an object of length $2$, and so on.

The $\infty$ symbol is just a formal symbol, and it says "If you reached this point - you've gone too far". It is not a number. But we can consider the case where we divide two infinities, e.g.

$$\lim_{n\to\infty}\frac{k\cdot n}{n+1} = \frac\infty\infty$$

But this limit can be calculated, it is in fact $k$.

It may contradict my previous statement, since those are both "infinite" numbers. However in this limit we compute the behavior of the ratios when taking bigger and bigger "lengths". We don't actually divide two infinities. So when we write $\frac\infty\infty$ we mean to say that this is a limit of the quotient of two sequences which grow larger and larger, but we cannot determine the exact result because we don't know what these sequences are, for example in the above example we can put $k=1$ to have the limit is $1$ and another pair of sequences where $k=2$ and the limit is $2$. Clearly $1\neq 2$. So we cannot determine a priori the result.

The case is similar for $\infty-\infty,\infty^0$ and $\infty\cdot 0$.

In addition to the answers you received, there are many books on the matter that you might want to consider. Here are some examples:

• Everything and More: A Compact History of Infinity, David Foster Wallace
• Infinity: Beyond the Beyond the Beyond, Lillian R. Lieber
• A Brief History of Infinity, Brian Clegg
• The Mathematics of Infinity: A Guide to Great Ideas, Theodore G. Faticoni
• Understanding Infinity, Anthony Gardiner
• In Search of Infinity, N.Ya. Vilenkin
• To Infinity and Beyond: A Cultural History of the Infinite, Eli Maor

Regards -A

In the context you were using, I'm pretty sure Mathematica considers $\infty$ to be the "extended real number" $+\infty$. The arithmetic of extended real numbers is the continuous extension of the operations on ordinary real numbers. The forms you wrote, $\infty / \infty$, $\infty - \infty$, and $\infty^0$ are all discontinuities of the respective operations, and are thus left undefined.

You got the result "Indeterminate" partly because Mathematica has a need to return a value anyways, and partly because such expressions often arise in the context of limit forms: when interpreted as limit forms instead of as arithmetic, they are all "indeterminate forms", so the word "indeterminate" is a reasonable choice for the return value.