Infinity is not a number

Infinity is not a "really really big number". It is not "the biggest number that exists". It's not the number you get when you divide by zero. There is no such thing as a finite infinity.

It's not equal to the volume of the universe, the density of a black hole, or the the IQ of God. Infinity isn't any of these things because it's not a number or quantity of any kind. Infinity is a concept.

Technically speaking, infinity is a transfinite number. (In this context, you're better off saying "omega-null".) Whatever concrete, finite number you pull out of your head, infinity is bigger. You can approach it (using a limit), but you won't ever reach it.

If you could treat infinity like a number, then you'd have to confront the fact that two times infinity is infinity, ten plus infinity is infinity, and infinity divided by 1048576 is infinity. Combining this with the factually incorrect idea that 1/0 = infinity, you could easily prove that all numbers are equal to each other and mathematics as a discipline is a bunch of self-contradictory nonsense.

This is in spite of the fact that your personal computer is still chugging along just fine in merry ignorance of your superior wisdom.

Why isn't infinity a number?
I haven't read any proofs here about why it actually isn't.

If the only reason it isn't is because it isn't defined, then wouldn't that suggest that the reason any number is a number is also simply the result of a definition? If that is so, wouldn't you satisfy mblase's statement, and you have: easily prove(n) that all numbers are equal to each other and mathematics as a discipline is a bunch of self-contradictory nonsense.

ie. if all numbers are purely defined concepts, then they are equatable simply as defined concepts.

Of course I am playing with language and reasoning but here is a more reasonable question:

1. Given some of the unique properties of the number 1, namely 1 divided by itself is still itself.
2. Given some of the unique properties of the number 0, namely 0 added to itself is still itself.
3. Why couldn't there be a number infinity which is itself whether it is added/mutiplied/divided to/by itself or any other number, just as 0 is itself when multiplied by another number.
4. Also, infinity does hold up to some rules of other numbers, infinity/infinity would still equal 1.

So basically, Infinity is a number, until proven guilty.

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Some people misunderstood the goal of this writeup, it was meant to illustrate some of the ways in which infinity might seem like a number in order to elicit accurate explanations of why it actually is not a number, rather than the previous writeups which mainly just stated that it was not a number. In any case ariels was nice enough to point out the the zero/infinity paradox which helps to explain why it is that x/0 does not equal infinity. For more on why it doesn't make sense to define infinity as a number look to the I can divide by zero node.

Think about what a number is--as I understand the foundations of mathematics, natural numbers are constructed from successive sets of the empty set. Thus, the number 0 corresponds to the empty set itself (represented as zero with a slash through it, or simply {}), 1 corresponds to the set of the empty set ({{}}), 2 to the set of that ({{{}}}), and so on. Now, for any natural number you name, I can write down the corresponding set*. I may need more time than I actually have before I die, but it is in principle possible. This is how we get natural numbers without merely assuming their existence.

This is not possible with infinity. Indeed, it is presumably for reasons such as this that Luitzen E. J. Brouwer and the intuitionists had problems with modern mathematics. The following quotation is taken from the Stanford Encyclopedia of Philosophy's entry on Constructive Mathematics (http://plato.stanford.edu/entries/mathematics-constructive/):

According to {Brouwer's} view and reading of history, classical logic was abstracted from the mathematics of finite sets and their subsets. ... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set theory, for which it is justly punished by the antinomies. It is not that such contradictions showed up that is surprising, but that they showed up at such a late stage of the game. (quoted in Kline 1972, p. 2001)

* Note that I'm not trying to claim that the epistemic justification for our belief in a mathematical object comes from the ability to write it down, only that if it is possible to use a well-defined method of notation to write something down, that that implies that it is in fact a clear concept. Similarly, the inability to find a consistent operation and clear method of communicating serves as weak, prima facie evidence that no clear concept answers to the description offered. For more on this, see JerboaKolinowski's excellent writeup in the I can divide by zero node.

The problem, really (apart from some early symptoms of MPAS), is definitions. Just what are you calling infinity, and what do you mean when you say number?

The thing is, "number" is such a basic concept to us, we forget it's actually an abstract creation of our minds. As such, we should probably try to pin down what we mean, when we discuss numbers. In a way, this is what mathematics is all about: inventing an abstract construct which seems to catpture some elementary pattern, defining it, and then studying how that construct behaves. Often, when we say "number", what we really mean is natural number, or rational number, or real number; sometimes even complex number. These are well-defined algebraic structures. Infinity is not a number in any of these senses; any element of these structures is evidently not "infinity". (Read the definitions and convice yourself, if you doubt this.)

So that settles it, right? Well, not quite. Mathematicians and physicists sometimes "add infinity" to their numbers (reals, or complex numbers, or whatever), depending on what they're going to do with them. There's nothing inherently wrong with this, so long as your definitions work... You don't "prove" that 1/0 is infinity; you can define it that way, and see if you get something workable. As it happens you don't! But then, mathematicians never were bothered with dividing by zero -- it's not meant to be defined. But now, start with the natural numbers, and add a new number X, and define that X+n=X (for any n). You might as well get used to calling it "infinity". Now see where that gets you...

Infinity is hardly something you run into in everyday life. The patterns abstracted by mathematicians to form their fields of study are, by and large, finite. But they might naturally lead to the study of infinity (e.g. the sets of natural and real numbers are clearly infinite); conversely, considering "infinity" as a "number" or "point" is often a mathematician's ploy to return to the realms of "finite" intuition, as exemplified by the Riemann sphere mapping.

More generally, the infinite is tackled by mathematicians in several ways. Try the concepts of series, limit, infinitesimal, and transfinites for starters. It's certainly there, in one guise or another...

Then, must we conclude that infinity is a number? It is, if you say so.

Well, the original noder is wrong. Infinite can be treated as a number, but only in special cases. Here are a few from the wonderful world of Calculus (for the purpose of this node x->y means as the number that x represents gets closer and closer to the number that y represents, see limit):

as x->infinity, x/x = 1

one divided by infinity is an infinitely small number, and can in specialized places be treated as zero. See 1/infinity IE: as x->infinity, 1/x=0.

In laymen's terms, as x gets closer and closer to infinity, 1/x gets closer and closer to zero. Whip out your calculator and try it out!

as x->infinity, x/(x+1) = one
This is because as x approaches infinity, the one becomes negligible.

infinite divided by one is infinity.
Infinite can also be the answer in several problems:

as x->0, 1/x equals infinity/negative infinity, depending on whether x is approaching zero from the positive (more than zero, x becomes smaller and smaller, ie 9,8,4,1,0.5) or the negative (less than zero, x becomes greater and greater, ie -9,-8,-4,-1,-0.5) side.

as x->0, cosx/sinx = infinite/negative infinite
This is because as x->0, cosx gets closer and closer to one, and sinx gets closer and closer to 0, and 1/x as x->0 equals infinity/negative infinity, once again depending on whether x is approaching 0 from the positive or negative side.

Join me next week, as I discuss more fascinating mathematical concepts!