Lines, Points, and Infinities

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?

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 
beads, doesn't it?  How about if instead you try to do the following: 
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 
both bags had the same number of beads.

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 natural 
  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 
instructor to help you read through this proof. Notice what a 
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 
about this interpretation. They are equal, but only in the very 
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
     http://mathforum.org/dr.math/problems/adams5.28.96.html   

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 
correct than yours.  If you are interested to read about this, it can 
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/