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 $\infty$?

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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 $B\cup\{x\}$).

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.

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.

There are many ways to interpret infinity, but without going into the details, let’s focus on why the operations you’re investigating fail.

The set of real numbers forms a field. A field is a mathematical structure that satisfies a few properties, mainly associativity, distributivity (of multiplication over addition), existence of additive and multiplicative identities, and so on. Fields also have an important property that they are closed under addition and multiplication–in other words, adding/multiplying two elements in a field gives you something in the field.

Any product/sum of real numbers will result in another real number. In a sense, there is no way to get infinity — infinity is not in the real field.

The operations of addition and multiplication are defined in the conventional way for elements in the real field. We can extend them to other fields, like the complex field, but we have to be more rigorous with their definitions.

If you want to treat infinity as a number (and there are ways to do this), you have to be very rigorous about how you define addition, multiplication, etc. As it turns out, you cannot just “append” infinity to to real number field and have the resulting structure still be a field! One implication of this is that our traditional notions of multiplication and addition (and consequently their inverses, subtraction and division) don’t work very well with infinity.

So in the traditional arithmetic, $\infty/\infty$ has no defined meaning. You can create an algebra and an arithmetic wherein it does, but W|A does not assume you are working in this realm.

Something that is important to note is that there is not just one “infinity”, but rather many, many infinities (in fact infinitely many infinities).

We say a set (collection of objects) is infinite if there is no 1-1 map into a set $\{1, 2, \ldots, n\}$ (1-1 just means don’t send any two things to the same place). So now we have a set that is not finite. An example of this would be the integers, the even integers, the real numbers (ones you would experience in calculus), the rational numbers, complex numbers, and the list goes on. Now we arrive at the question of whether these infinities are the same.

In math, we say two sets $S$ and $K$ are the same size if there is a bijective map between the two, i.e. if we can assign to every $s$, a value $k$, such that every $s$ has a buddy $k$. To illustrate this, let’s look at the even integers vs. all the integers. Are they the same size?

Well, let’s define a map $f : \mathbb{Z} \to 2\mathbb{Z}$, sending $a \mapsto 2a$. This will assign to every integer an even integer, and every integer is “hit” by this map, so we can say that the two sets have the same size.

Showing two sets don’t have the same size is a little more difficult. To see that the real numbers are bigger than the integers you can look up “Cantor Diagonalization”.

You can also look up “cardinals and ordinals” to find information on different set sizes.

A turning point in my understanding of infinity occurred while reading the book titled “Number: the language of science” by Tobias Dantzig. The following passage is on page 231:

“This, too, may have been an early idea of Cantor. But he proved conclusively that here too our intuition leads us astray. The infinite manifold of two or three dimensions, the mathematical beings which depend on a number of variables greater even than three, any number in fact, still have no greater power than the linear continuum. Nay, even could we conceive of a variable being whose state at any instant depended on an infinite number of independent variables, a being which “lived” in a world of a denumerable infinite of dimensions, the totality of such beings would still have a power not greater than that of the linear continuum not greater than a segment one inch long.”

What an amazing idea! Here is the wikipedia version:
“Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.”

This is the way I perceive it. Between any two moments in time there are an infinite number of moments, therefore we live an eternity from moment to moment.

Infinity in the mathematical context is really just a shorthand. All usages of $\infty$ can be rephrased without it! Often that rephrasing is the better choice: It is often more meaningful to say “increases without bound” than saying “approaches infinity”. I have noticed people, especially physicists, who tend to like wordings avoiding explicit use of $\infty$. We should follow them!

I am not much of a mathematician, but I kind of think of infinity as a behavior of increasing without bound at a certain rate rather than a number. That’s why I think $\infty \div \infty$ is an undetermined value, you got two entities that keep increasing without bound at different rates so you don’t know which one is larger. I could be wrong, but this is my understanding though.

First of all, infinity is not a number, it’s more like a concept. That’s why you can never put infinity on any side of an equation or put it in any mathematical operation.
For example the following expression is not defined: $ n = \infty$. It’s like writing something like $n=apple$.
It has no mathematical significance, and that’s why also any of the following expression aren’t defined: $\infty^0$, $ \infty \over \infty$ and so on.

A lot of people also think that $1/0 = \infty$ but guess what? that is also wrong.
Why? apart from the reasons I just described , notice that $2/0 = \infty$ and so , one can always say: $2/0 = 1/0$ leading to $2=1$ which is absurd.

On a final note, a mathematical expression can only tend to infinity. When we say that a mathematical expression tends to infinity, we usually mean that for every number we choose, this mathematical expression can be evaluated to be greater.

$$\infty=\lim_{x\to 0}\, \frac{1}{x}$$

It is the unreachable end as we move along increasing side on positive real axis.