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### Lines, Points, and Infinities

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Date: 09/01/2001 at 01:36:55
From: Graham
Subject: Lines and infinity

My geometry teacher told us to imagine that we had two line segments,
one double the length of the other. The short line segment, like all
lines, is made of and contains an infinite number of points; the
longer one, twice as long, has twice as many points, or 2 times
infinity (or the short segment has 1/2 infinity, the longer infinity).

My friend and I believe that he is mistaken. Here is our case:

- points, lines, planes, and infinity cannot exist in this universe,
but are only abstractions of our minds, a form of mythology to
explain our world.

- infinity is an abstraction, so it is not real; it cannot be halved
or doubled or what-have-you; infinity is infinity, and that is
that.

- therefore, a line - whether segment, ray, or normal line - of any
length at all, even one infinitely long, contains EXACTLY an
infinite number of points in it.

Am I correct in my reasoning?
```

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Date: 09/01/2001 at 10:20:29
From: Doctor Jordi
Subject: Re: Lines and infinity

Hello, Graham. Thanks for writing.

Your reasoning seems to strongly parallel the reasoning behind
Cantor's treatise on infinite sets. I would side with your argument
that the "numbers" of points in both lines (I use quotes because
infinity is not a number in the usual sense) are equal: infinity.

However, let's not be hasty and jump to conclusions before we explore
more what kinds of infinities there could be and if indeed there is
one and only one concept of infinity that we need.

Let us associate these two lines, one of length twice as great as the
other, with the real number. Say, to every point of line A, the short
one, we associate a real number between 0 and 1, inclusive. For line
B, let us associate to every point on this line with a real number
between 0 and 2, inclusive. The question now becomes: "Are there more
real numbers between 0 and 1 or between 0 and 2?" Before we answer, we
had better explain what "more" means in this case, since both sets are
infinite. How do we compare the size of infinite sets?

Simple: we count.

Think about what counting means. I have two bags of beads of various
shapes, sizes, and densities, and I ask you, "which bag contains more
beads?"  In this case, it is simple to answer because you know that
there is a finite number of beads in each bag. You could grab bag A
and take out beads one by one and place them on a tray. Each time you
take out a bead you say a natural number out loud, in sequential
order. "One, two, three, four, ..."  Eventually you will run out of
beads, because they will all be on the tray. To each bead you have
assigned a natural number. The number of beads is the last natural
number you assigned to the last bead in bag A. Then you could repeat
the process with bag B, starting again from 1, and find the largest
natural number you can get to before running out of beads.

In other words, counting in this sense involves putting a set of
objects in a one-to-one correspondence with the set of natural
numbers.

This seems like a bit of extra work to know which bag contains more
with one hand you take out a bead from bag A and with the other hand
at the same time take a bead out of bag B. Discard them together onto
the tray. Repeat the process. Eventually, you will run out of beads in
one of the bags, or perhaps in both at the same time. If you run out
of beads in bag A first, then you know bag B contains more elements.
If you run out of beads in both bags at the same time, then you know

That is to say, to compare the size of two sets it suffices to attempt
to form a one-to-one correspondence between the elements of the sets.
If such a correspondence exists, then both sets are the same "size"
(the technical word is "cardinality").

So let us rephrase our question once more: What is the cardinality of
the set of real numbers between 0 and 1? Is this cardinality less
than, greater than, or equal to the cardinality of real numbers
between 0 and 2?  Just for dramatic impact, let me give the answer in
the form of a

THEOREM: The cardinality of any two intervals of real numbers is the
same.  Moreover, the cardinality of any interval is equal to the
cardinality of the entire set of real numbers.

Proof: An interval denoted as [u,v] is the set of all real numbers
between u and v, inclusive (we assume u is less than v).  So in
order to prove that the cardinality of any such two intervals is
equal, we only need to give a one-to-one correspondence between any
two such intervals, [a,b] and [u,v].  To this effect, consider the
function f(x) = (v-u)/(b-a)*(x-a) + u, for any x in the interval
[a,b].  This function takes in any value between a and b and pairs
it off with exactly one value between u and v.  As a concrete
example, if [a,b] = [0,1] and [u,v] = [0,2] then we have

f(x) = (2-0)/(1-0)*(x-0) + 0
= 2x

Thus, take any number in [0,1] and double it.  You will have then
paired it off with exactly one number in [0,2].

To prove the second statement of the theorem, we will again
construct a one-to-one correspondance between the set of real
numbers and a specific interval, (-1, 1) (the use of parenthesis
instead of brackets to denote the interval means that the endpoints,
-1 and 1, do not form part of the set of real numbers we are
considering). Consider the function

x
f(x) = -------      for any real number x, where |x| is the
|x| + 1       absolute value of x

This function takes in any real number x and spits out a number
between -1 and 1, because |x| < 1 + |x|.  Thus, since the entire set
of real numbers can be  paired off with the set of real numbers in
the interval (-1, 1), and since the the set of real numbers in any
interval can be paired off  with the set of real numbers in any
other interval, the set of real  numbers has the same cardinality as
any interval of real numbers.

Read carefully through that proof and make sure you understand every
point. I hope that you are familiar with the function concept and feel
comfortable using it. If not, perhaps you may wish to ask your
geometrical interpretation of the final conclusion could be: a line of
infinite length (the real number line) has the same number of points
as any finite line.

A couple of remarks: Think carefully about what we have proven here.
We have proven that there is a one-to-one correspondence between the
real numbers in [0,1] and those in [0,2]. It makes sense, somehow, to
thus conclude that "the two infinities are equal," but be careful
restricted sense we have given here. Further, I should warn you that
not ALL infinities are equal, under this interpretation. To see more
clearly what I mean, take a look at the following links from our
archives:

Sets Containing an Infinite Number of Members
http://mathforum.org/dr.math/problems/kate2.3.98.html

Infinite Sets
http://mathforum.org/dr.math/problems/lee7.17.97.html

Infinite and Transfinite Numbers

There are other interpretations possible. An important example is one
in which we can form an entire arithmetic of infinite (and
infinitesimal) numbers, not unlike the arithmetic with real numbers,
where it would make more sense to claim the number of points in [0,1]
is half of that in [0,2]. This can be done in a very specific subject
called nonstandard analysis. I am not sure if your instructor intended
to use this interpretation, but if he did, then his claim may be more
be found in our FAQ:

Nonstandard Analysis and the Hyperreals
http://mathforum.org/dr.math/faq/analysis_hyperreals.html

This would mean that the disagreement between you and your instructor
might have arisen because you were each playing according to different
rules.  Or, as Thoreau wrote: "If a man does not keep pace with his
companions, perhaps it is because he hears a different drummer."

If you would like to follow up on this e-mail, by all means do.

- Doctor Jordi, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
College Analysis
College Logic
High School Analysis
High School Logic
High School Sets

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