Misconception #1: Infinity is just a "funnylooking number"
Now, I concede that ∞ is funnylooking. It looks like an 8 written
by a sideways mathematician (or a sideways 8 written by a normal
mathematician). It can also be used for such questionable practices as the
following ASCII figure (I mean, Unicode figure):
~o
/∞\

/ \
Which I think looks better than
K5ARP's
sideways "B"
in place of the ∞, don't you think? (And that's a swimsuit, not
something else, you pervs.)
But aside from the funnylooking part, "infinity" is not a number.
Let me repeat that. "Infinity" is not a number!
That is to say, for any real number x, things like ∞/∞,
x/∞, ∞/x, are all nonsensical. When a mathematician says 1/∞
= 0, you have to understand it in the right context (explained in a moment).
When a human being says 1/∞ = 0, he's talking nonsense; ignore him.
What context, you ask? First of all, "1/∞ = 0" is just a
mathematical shorthand; it is not saying that if you divide 1 by
this strange beast (or, lazy beast) called ∞, it magically
turns into 0. Remember, ∞ is not a number. What the
mathematician means when he (uh, it) says that 1/∞ = 0, is that
if you take 1 and divide it by a very large number x, it will be
"close to" 0, and that the larger the number, the closer 1/x will be to 0.
Note that it is merely "close to 0", but not actually "equal to 0". The
"proper" way to write this is:
lim_{x→∞} 1/x = 0
In mathspeak, this is read "the limit of 1/x, as x approaches infinity,
is 0".
"Aaack, you just fooled me!" I hear you scream. "What do you mean, x
approaches infinity? You just said infinity wasn't a number!"
Calm down. When mathematicians say "x approaches infinity", or write
"x→∞", they mean "x grows arbitrarily large". And when they say
that the limit of something is 0, they mean that it can go as close to
0 as you want it to, but it may or may not be actually equal to
0. They do not mean that x "becomes" ∞ (it cannot, since
∞ is not a number to begin with), or that 1/x becomes equal to 0 (since
1 divided by any number is never equal to 0). When humans (or math students)
say or write this, however, they are just talking nonsense; ignore them.
Oh, and did I mention that ∞ is not a number?
Misconception #2: Wash, rinse, repeat (forever)
This is often the highschool teachers' copout explanation of infinity.
"Oh, you just repeat this forever and you get infinity."
Bzzzt. No, you do not. You get nothing. Not even zero.
Nothing. Nonsense. Nada. "Repeat this forever" is nonsense. Although you
could repeat something forever, you won't get anything out of it;
you'd just get really, really, tired.
Here's a typical highschool description of the set of natural numbers:
"You start with the empty set, { }, and you add 0 to it to get {0}, and then
you add 1 to it to get {0,1}, and then you add 2 and get {0,1,2}, and so on,
and you repeat this forever, and you get N, the set of all natural
numbers."
This description is not only inaccurate, it is just plain
wrong on some points. You do not get anything if you
repeat this forever; sure, your set will get bigger and bigger, but that's
about it. Since you are repeating forever, by definition there is no result
yet, and there never will be any result, since you never stop.
So, class, what do you get if you repeat this forever? Repeat after me:
"Nothing. Nothing. Nothing. Nothing. Nothing. ..." (repeat this
forever)
Mathematically speaking, this description is wrong because:
 The set of natural numbers is not "built up" by repeating some
operation forever. You can never reach infinity by repeating something finite.
That's why it's called infinity, duh.
 Even if it were possible to repeat something forever, you'd still
not get the set of natural numbers. If I built a machine that alternately
prints 1 and 0 every second, what will it print after infinite time? If you
say 1, I can equally argue it should be 0. If you say 0, I can equally argue
it should be 1. Or you could say it's both, but "both" is not a number. The
real answer: there is no such thing as getting a final result out of
"repeating forever". "Forever" means there is no final result. So please get
that silly idea out of your head already.
The set of all natural numbers
"Then how do you get the set of natural numbers?!" I hear
you ask.
The answer: we get it by definition. For you math buffs, this
definition is called the Axiom of Infinity. (And it is called that, precisely
because it defines what infinity is.) Don't get fooled by the scary name of
this Axiom; it is really rather simple. All it says is this: there exists a
set (conventionally called N), which contains the number 0, and if N contains
some number x, then N also contains (x+1).
Let's unpack this sentence a bit. First, it says N contains 0. OK, so N
looks like {0, ...}. But since it contains 0, it must also contain (0+1),
which is what we call '1'. So N must look like: {0,1, ...}. Furthermore,
since it contains '1', then it must also contain (1+1), which is what we call
'2'; so it looks like {0,1,2, ...}. By the same reasoning, it must also
contain what we call '3', so it would look like {0,1,2,3, ...}. And so
forth.
"Hold on a second there!" I hear you say. "Isn't this just another case of
`repeat forever' that you just debunked??"
No. Please notice the subtle, but very important difference:
we are not building anything by repeating some operation forever.
We are simply defining N to contain every natural number there is.
We are not building N piecemeal; we're not saying "add (x+1) to N". We're
saying that for every number x, (x+1) is already in N. We're not
adding numbers into N to make it N; every number is already in N.
The whole deal with the (x+1) in the Axiom of Infinity is so that we can
describe N fully in a finite amount of space and time. Otherwise, we would
need an infinitely long paper to list all the members of N, and by definition,
we cannot ever finish writing this list, so it is not a proper
definition of N, and will never be.
N, the set of natural numbers, contains every number there is by
definition. It is not "attained to" by doing some "repeat this forever"
nonsense. You cannot attain to infinity; you need to be
given an alreadycompleted infinity by a Higher Power. In this case,
the Higher Power is the Axiom of Infinity.
Misconception #3: Infinity is nonsense because it doesn't follow the
rules of arithmetic
Or, as some people would say:
I was told that 1/∞ = 0. If I multiply both sides by
∞, I get 1 = 0*∞. But since 0*anything = 0, I get 1=0. Argh, my
head hurts!
I'm not surprised your head hurts. Besides the fact that infinity is not a
number (see misconception #1), this misconception also underlines a faulty
assumption people have: that infinite quantities obey the same rules of
arithmetic as finite quantities. They do not. If they did, we could prove
that every number is 0 using the above argument: let n be any
nonzero number. Since 1/∞ = 0, we can multiply both sides by
n, and get n/∞ = 0. But that means that
n = 0*∞, so we conclude that n=0.
Obviously, this is ridiculous. As I said before, and I'll say again,
∞ is not a number.
But there are such things as infinite quantities, and it is
in fact possible to define consistent arithmetic on them. Just don't
expect them to behave like finite quantities. (That's why they're called
infinite quantities!)
Comparing infinite quantities
Infinite sets cannot be compared by counting the number of elements in
them, and then comparing the totals. You cannot ever finish counting elements
in an infinite set, because it's infinite! However, it is possible to
compare them without ever counting them. Here's how.
Suppose we were to turn back the clock, to the time when we were 2 years
old and didn't know how to count up to 5 yet. How would we know if we had the
same number of fingers on either hand? We couldn't count the fingers on each
hand, and compare the results; because we couldn't count that high. But we
could put our hands together, finger to finger, and find that there
are no extra or missing fingers that didn't have a matching finger on the
other hand. We could then conclude that there are as many fingers on our left
hand as our right hand, without knowing how to count how many there
are!
Mathematicians call this establishing a 1to1 correspondence. In
this case, we're establishing a 1to1 correspondence between lefthand
fingers and righthand fingers. If there's a 1to1 correspondence between two
sets A and B, we say that they have the same cardinality.
("Cardinality" is just mathspeak for "size".) For infinite sets, we may not
be able to count the number of elements in them; but we sure can tell if two
of them had the same cardinality (size) by seeing if we can make a 1to1
correspondence between their elements. (This, of course, also works perfectly
fine with finite sets; so we're not using some "crippled" method of comparison
here.)
Here's an example. Take the set of all natural numbers, N, and throw away
the odd numbers. Call the result E, the set of even numbers. Question: how
many elements does E have? Surely E must only have half the number of elements
as N, since we threw away the other half? Not really. Consider this
correspondence: for every element x in N, we map it to x*2. Since x*2 is even,
x*2 is an element of E. Every number in N is mapped to exactly one element in
E, and every element in E has exactly one number in N mapped to it. The
fingers match! This means that E and N in fact have the same size.
(Note: we only had to find one possible 1to1 correspondence.
Just because some correspondences don't work doesn't mean the sets aren't
equal in size. Just because the 2yearold didn't manage to match his index
fingers together doesn't mean one of his hands is missing an index finger, as
long as there is some way to match them all up correctly.)
"Wait!" I hear you say. "Isn't this a contradiction? We just removed some
elements from a set, shouldn't it be smaller now?"
Well, that's why it's called an infinite set. If your computer is
connected to a battery that stores enough power to run for 10 hours, then
after each hour, the battery has 1 less hour's worth of power left. After 10
hours, it runs out of power. This is your typical finite set: remove something
from it, and it contains less than before. But if you plugged your computer
into the power outlet, and it continues to consume the same amount of power
per hour as before, the power never runs out! How can this be??
Because the power outlet is connected to your local power plant, which is
producing more power as your computer consumes it. This is your
infinite set: removing something from it doesn't necessarily make it smaller.
The power plant is an "infinite battery"; consuming power from it doesn't make
it "contain less power".
So when dealing with infinite quantities, don't blindly assume that the
rules of finite arithmetic automatically apply to them. They don't. And just
because they don't, doesn't make them contradictory.
Misconception #4: There are twice as many integers as there are natural
numbers
We've already seen that there are just as many even numbers as there are
odd numbers. Now I'm going to show you something even more surprising: the
positive and negative integers, taken together, are still not any
bigger the set of all natural numbers!
How? Well, consider the set of integers, Z. For every integer z in Z, if z
is positive, we map it to (z*2)+1, which is a positive odd number. If z is
negative, we map it to z*2, which is a positive even number. And if z is 0,
it stays 0. Note that every odd number is covered, and so is every even
number, including 0. Miracle! Did you see what just happened? What looked like
two "infinite endpoints" in Z has collapsed into a single endpoint, and we are
back to N, the set of natural numbers, yet again!
"OK", I hear you say. "But what about fractions of integers?"
Good question. Fractions of integers, which mathematicians call "rationals"
or "rational numbers", are numbers of the form p/q, where p and q are
integers. So 1/2, 1/3, 1/4, 2/5, etc., are all rationals. The set of rationals
is conventionally called Q (for quotient).
Q is an interesting set that has a property called density. That
means that between every two rational numbers, you can always find another
rational in between. For example, between 0 and 1, you have 1/2, and between
1/2 and 1 you have 3/4, etc.. It's easy to see that between every two numbers,
there is an endless supply of rationals. This is why we say Q is
dense; no matter how many times you magnify it, you will still see it
jampacked with rationals.
Now the milliondollar question: is Q larger than N?
Misconception #5: Since the set of rationals is dense, there
must be more rationals than integers
On the surface, it sure looks like this is the case. After all, not only do
the rationals stretch infinitely to either side of 0, there are also an
infinite number of them between every given two. There's no way this isn't
bigger than the set of natural numbers!
Well, don't be fooled by the appearance of Q. (That applies in Star Trek,
and it applies here, too.) Just like Z, the set of integers, Q is just
arranged funny. If we straighten it out the order of its elements, it'll look
a lot less scary.
"OK, you've got to be kidding me," I hear you retort. "There is
no way you're going to rearrange a dense set into a
nondense one!"
Just wait till you see it. First, to make it less confusing, let's restrict
ourselves to Q^{+}, the set of positive rationals. Note that it is
still dense, so there's no sleight of hand here. Now, Q^{+} consists
of fractions of the form p/q, where p and q are positive integers. So we make
a table, with the columns representing p, and the rows representing q, like
this:
1/1 2/1 3/1 4/1 ...
1/2 2/2 3/2 4/2 ...
1/3 2/3 3/3 4/3 ...
...
OK, some of these entries are redundant, since 2/2 is the same as 3/3,
etc., but the point is that this table contains every possible rational in
Q^{+}. Now, we cut this table up into diagonals. The first
diagonal is 1/1; the second is 1/2 and 2/1; the third is 1/3, 2/2, 3/1; etc..
Since each of these diagonals are of finite length, we can glue them together,
end to end, to form a list that goes like this: 1/1, 1/2, 2/1, 1/3, 2/2, 3/1,
... . Of course, we can remove the redundant elements from this list,
but the point is that we can arrange all the elements of
Q^{+} into a linear, nondense list.
You can probably guess what's coming next. That's right, we label each
element in this list, starting with 0, 1, 2, ..., etc.. It's the set of
natural numbers again! So we find out that Q^{+} is actually the same
size as N. But we can do exactly the same thing with Q^{}, the set of
negative rationals. So Q^{} is also the same size as N.
Now the kicker: when we labelled the elements of Q^{+}, we could've
used even numbers instead; and when we labelled the elements of
Q^{}, we could've used odd numbers. Put them together, and
BANG! we get N, yet again! So then, there are only as many rationals as there
are integers, even though the rationals are dense. Betcha they didn't tell you
this in highschool!
You're probably starting to notice a trend here. An awful lot of sets seem
to be the same size as N. In fact, N is so special, that mathematicians have
come up with a name for the size of N. We say that the cardinality of N is
ℵ_{0}. That's pronounced "aleph null" or "aleph zero", by the
way; not "squiggly egg". Since Z and Q are also the same size, they
all have cardinality ℵ_{0}.
"Why not just call it 'infinity'?" you ask.