Is infinity a number?

Question

Is infinity a number? Why or why not?

Some commentary:
I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school — but a difficult one to answer in an intelligent manner. I'm hoping to see a combination of strong citations and solid reasoning in the answers.

Answer 1

It comes down to the definition of "number," as well as the definition of "infinity." Personally I don't think it's worth having an opinion on this subject; there are more precise words than "number" and "infinity" in mathematics. Historically the word "number" has come to mean an increasingly general list of things:

• a positive integer
• an integer
• a rational number
• a constructible number (say, to the ancient Greeks)
• a real number
• a complex number
• a cardinal number
• an ordinal number
• a surreal number...

The word "infinity" has also come to mean an increasingly general list of things: it might refer to

• a countably infinite set
• an uncountably infinite set
• the point at infinity in projective geometry
• one of the two new points in the two-point compactification of the real numbers
• the new point in the one-point compactification of the complex numbers, also known as the Riemann sphere
• an infinite cardinal number
• an infinite ordinal number...

Some of these meanings are compatible, as the above list demonstrates. But again, there are more precise words than "number" and "infinity" in mathematics, and if you want to get anywhere you should learn what those words are instead.

Here are some of those more precise words.

• A set is a formalization of the intuitive notion of a bag of objects, and we can talk about finite or infinite sets. For example, $\{ 1, 2, 3 \}$ is a finite set, whereas the set of natural numbers is an infinite set. One can do arithmetic with sets in a way that leads to the arithmetic of the natural numbers: for example, taking the disjoint union corresponds to addition, and taking the Cartesian product corresponds to multiplication. These ideas lead to the arithmetic of the cardinal numbers, and similar ideas lead to the arithmetic of the ordinal numbers.
• A ring is a formalization of the intuitive notion of a set of things you can add and multiply, so in some sense one can regard elements of rings as "generalized numbers" (but note that not every generalization I listed above can be interpreted in this way). When certain people say that "infinity is not a number," what they mean is that you can't adjoin an element called $\infty$ to the ring $\mathbb{R}$ of real numbers such that addition and multiplication do what you want them to do, the basic problem being that $\infty - \infty$ can't be consistently defined to satisfy the other rules of arithmetic if you also want it to be true that $n + \infty = \infty$ for any finite $n$.
• A field is a commutative ring in which it's also possible to divide by nonzero elements. Some people would like to say that $\frac{1}{0} = \infty$, but by mathematical convention the element $0$ never has a multiplicative inverse in a field, the basic problem being that $0 \cdot \infty$ can't be consistently defined to satisfy the other rules of arithmetic. However, one can make sense of the expression $\frac{1}{0}$ in projective geometry; it describes the point at infinity on the projective line.
• A topological space is an abstract setting for ideas like nearness and taking limits. Sometimes we don't want to view $\mathbb{R}$ as a ring, but as a space (the real number line), and we can talk about embedding this space into a larger space where more limits exist: this is known as compactification, and is an extremely useful tool in mathematics and physics. For example, we would like to say that the sequence $1, 2, 3, ... $ has limit $\infty$ in some sense, and we can do this compactifying $\mathbb{R}$.

Answer 2

The word "number" carries some baggage with it. If $a$, $b$, and $c$ are numbers, and $a+c=b+c$, one expects $a=b$. For infinity, that doesn't work; under any reasonable interpretation, $1+\infty=2+\infty$, but $1\ne2$. So while for some purposes it is useful to treat infinity as if it were a number, it is important to remember that it won't always act the way you've become accustomed to expect a number to act.

Answer 3

To simplify (with respect to Qiaochu's answer), most people think of: - 'number' as an integer or a real (a count or length), with their usual properties of addition and multiplication. - 'infinity' as some vague unreachable bigger than any integer or real, so that it doesn't follow all the same rules as integers or reals.

So in common English, 'infinity' is not a 'number'.

But of course in math, there are technical methods for dealing with a particularly defined 'infinity' so that it can be manipulated in the same sentence as an integer or real, so that one notices that though it doesn't share all properties with things called 'numbers', it shares some.

Even then, one normally does not call 'infinity' a 'number', but it does share some properties with them and can (sometimes be manipulated with them). So one might say that 'infinity' can be treated as a number sometimes.

That is, 'X is a Y' is not such a straightforward thing to answer.

Answer 4

To answer your question directly: no, infinity is not a number. That is, for most people in most situations most of the time, infinity is not included in the set of numbers.

Qiaochu gave a very good answer and I agree with his opinion parts. As Qiaochu pointed out, it depends largely on which definitions you choose to accept, or use, at any given time. Sometimes it's useful for infinity to be a number, other times it isn't. When Qiaochu talks about rings, sets, fields and spaces, and when I say "sometimes yes, sometimes no", what we're doing is referencing a group of axiom sets, or rule sets, which has different definitions and rules as to what numbers and infinity are.

If you're interested, this topic bleeds into a discussion about the history of math, mostly the last 200 years. Basically, the idea that any given definition or rule is "absolutely right" or "absolutely wrong" was discarded in favor of a more axiomatic approach. That is, things may be one way one day, and another way another day; and we, as mathematicians, are ok with that.

For a quick concrete example, ask yourself, "Is 2.5 a number?" If you're counting people, it is often absurd to consider 2.5 a number (thus the joke in the T.V. show). On the other hand, if you're measuring mass, it may be absurd to ignore 2.5 as a number. So the question, "Is 2.5 a number?" has no one answer all the time. In the same manner, the question "Is infinity a number?" has no one answer all the time.

Some interesting highlights on the history of math and infinity: -Alice's Adventures in Wonderland, by Lewis Carroll, is supposed to be a sarcastic take on the emergence of modern math, notably Non-Euclidean Geometry and Abstract Algebra. Carroll insists that math has was fine for the last 2000 years, and it doesn't need to be adjusted. He portrays emerging math concepts as absurd nonsense in wonderland. -Georg Cantor rocked the boat with his work on infinity.

As far as describing it in an intelligent way, just say that infinity is not a number because infinity is a meta word not in the set but used to describe the set. Just as the words "unbounded" and "non-empty" are (usually) not considered as numbers, infinity is (often) not considered as a number.

Answer 5

Infinity is not a number, but some things that can reasonably be called numbers are infinite. This includes cardinal and ordinal numbers of set theory and infinite non-standard real numbers, and various other things.

There are various different things called infinity. There's the infinity of complex analysis, and there are the $\pm\infty$ encountered in calculus. There is the "infinity" that is the value of Dirac's delta function, which changes when multiplied by a real number. And there are many others.

Answer 6

I think this more a philosophical than a mathematical question. Since I am a philosopher, I will answer it. I believe "infinity" does not name a number (nor does it name a class of numbers, or anything like that).

Arguments that infinity isn't a number:

1. We can't conceive of a number that is larger than every finite number; however large an amount you imagine, there is a natural number (or a real number) larger than that.
2. Arithmetical operations don't apply to infinity. E.g., if we say "$\infty + 1 = \infty$", this seems to imply 1 = 0. (It also suggests there is a number that is one larger than itself.) If we say "$1/\infty = 0$", then also $2/\infty = 0$, and then it seems that 1=2. But the ability to add, subtract, etc., with something is essential to the concept of a number.

3. Greater-than, less-than, and equal-to relations don't apply in the same way to infinite "numbers" as they do to finite numbers. An essential feature of numbers and the greater-than relation is that they satisfy this principle:

   (P) Adding to a number makes it larger.

   Infinite cardinal "numbers" violate this, because when one adds to an infinite set (without taking anything away), the resulting set may not be larger than the original set. (I.e., when A, B are disjoint and A is infinite, $A \cup B$ may not be larger than A.)

Counter-argument: Some people (Frege, Russell, and others following Cantor) consider there to be infinite numbers, because:

1. They have a theory of the nature of numbers: they think a number is a certain kind of set.
2. It is possible to describe certain sets, which are recognizably of the same kind as the sets we identify with natural numbers, except that they would have to correspond to infinite numbers.

In Russell's version of this, for example, the number "1" is the set of all sets that are equinumerous with {0}, "2" is the set of all sets that are equinumerous with {0,1}, etc. Then what about the set of all sets that are equinumerous with {0,1,2,3,...}? Looks like that should be a number too. But it would be an infinite number (namely, $\aleph_0$).

Reply: That argument depends on the assumption that the natural numbers are sets of the kind described. A better theory is that the natural numbers are plurally-instantiated properties (properties that, when exemplified, are exemplified by multiple things).

Answer 7

If you want to take time to listen to something about the history and use of "infinity" then the BBC Radio 4 programme "In Our Time" ran a programme on this subject in 2003 with Ian Stewart, Professor of Mathematics at the University of Warwick; Robert Kaplan, co-founder of The Math Circle at Harvard University and author of The Art of the Infinite: Our Lost Language of Numbers; and Sarah Rees, Reader in Pure Mathematics at the University of Newcastle all contributing. This may not quite go into the depth and rigour that you are after, bearing in mind the "general" audience it is aimed at, but it may well be of interest still.

I'm not sure of the availability of this to people that reside outside the UK however, so this may not be of use to the OP.