Lines, Points, and InfinitiesDate: 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 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 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/ |
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