TITLE: Is infinity a number?
QUESTION [114 upvotes]: 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.

REPLY [3 votes]: 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:

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.
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.
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:

They have a theory of the nature of numbers: they think a number is a certain kind of set.
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).