On 7/20/2013 4:27 PM, Warp wrote:
> So it's a number that's not any kind of number? It's not an integer,
> it's not rational, it's not irrational, it's not transcendental, and it
> doesn't follow any of the rules of any of the other sets (eg. if you
> add 1 to any number, you get a new number that's larger than the
> original; or if you multiply any number by 0 you get 0.)  All mathematical
> operators and functions work completely different for it than for any
> other numbers (moreover, most of them aren't even well-defined for
> infinity.)

Before I say anything, it's important to point out that I don't find the 
term "number" to be particularly mathematically meaningful in the first 
place.  It's probably best to just be explicit to say "integer", "real", 
"cardinal number" etc., and not bother too much about what counts as a 
"true number" and what doesn't.  That said, if you're going to take some 
sort of definition as to what counts as a "number", then there are some 
good reasons that you might want to include some notions of infinity as 
numbers.

What you say above is true for the particular type of infinity you seem 
to be thinking of, but it is *not* true in general for other definitions 
of infinity.  I'll address it point by point:

Q) It's not an integer, it's not rational, it's not irrational, it's not 
transcendental

A) If this is your criteria then you've basically said from the 
beginning that infinity isn't a number by definition -- no matter how 
much like a number it behaves.  You can certainly take this approach, 
but you'll need to accept the existence of "number-like things which are 
not numbers".  This is completely valid, in which case I don't think we 
really disagree in any substantial way but just draw the line between 
"number" and "number-like thing" is different places.

Q) it doesn't follow any of the rules of any of the other sets (eg. if 
you add 1 to any number, you get a new number that's larger than the 
original; or if you multiply any number by 0 you get 0.)

A) This is not true in general.  There are definitions of infinity for 
which infinity+1 > infinity, and for which infinity*0 = 0.  And besides, 
there are similar situations with "normal" numbers anyway.  For instance 
in the reals you can calculate 1/x for every number x except 0, and 
something like x+1 > x doesn't makes sense with complex numbers.  Heck, 
in the integers x/y only makes sense if y if a factor of x.  "Normal" 
numbers have arithmetic exceptions too, and it doesn't prevent you from 
thinking of them as genuine numbers.

Q) All mathematical operators and functions work completely different 
for it than for any other numbers (moreover, most of them aren't even 
well-defined for infinity.)

A) I actually find the change in how things work by including infinite 
numbers to be *less* than the difference between the integers and the 
reals.  Saying "All mathematical operators and functions work completely 
different for it than for any other numbers" simply isn't true in 
general.  It's certainly true for *some* definitions of infinity, but 
for other definitions you get perfectly well defined addition and 
multiplication, and for a few you even get commutative addition and 
multiplication, as well as division, subtraction, etc.  Heck there are 
some with these properties where you even maintain that either a <= b or 
b <= a, which isn't even true of the complex numbers!

I mean, obviously you need to draw the line somewhere between what you 
call a number, and what behaves differently enough that the term 
"number" no longer makes sense, but I think it's pretty reasonable to 
consider some (but all) definitions of infinity as sufficiently 
number-like to be genuine numbers.  I also think that it's fine to call 
complex numbers, quaternions, elements in cyclic groups etc. as numbers 
too, so maybe I just take a pretty broad stance as to what I think of as 
a number.


> Infinity is not a number. It's just a concept that's used to describe
> a more abstract notion. You can use it a bit like if it were a number
> when dealing with things like limits, but even then it's just a shortcut
> notation.

This is math we're talking about, everything is abstract in some sense. 
  I agree with limits that it's a notational shortcut, but there are 
many many other ways of defining infinity quantities, and some of them 
behave quite a bit like numbers.

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