How can points that have length zero result in a line segment with finite length?

Question

I have been told that a line segment is a set of points. How can even infinitely many point, each of length zero, can make a line of positive length?

Edit: As an undergraduate I assumed it was due to having uncountably many points. But the Cantor set has uncountably many elements and it has measure $0$.

So having uncountably many points on a line is not sufficient for the measure to be positive.

My question was: what else is needed? It appears from the answers I've seen that the additional thing needed is the topology and/or the sigma algebra within which the points are set.

My thanks to those who have helped me figure out where to look for full answers to my question.

Answer 1

This may seem rather a strange thing to say, but I don't think it's helpful to think of lines as made up of points: the "lininess" of a line is an inherent property that points don't have, so it has some extra qualities that points don't, such as length.

The real numbers are basically the answer to the question "How can I augment the set of rational numbers so that I don't have to worry about whether limits that ought to exist really do exist?", from which one can then do calculus. One can wheel out $\sqrt{2}$, $\pi$ and so on if one so desires as an obvious example of a point where one needs this.

Perhaps a more helpful introduction of the real numbers is to say "I want to know how far I am along this line." You then say "Am I halfway?" "Am I a quarter of the way?" "Am I 3/8ths of the way?", and so on. This gives you a way of producing binary expansions using closed intervals, and you can then introduce the idea of asking infinitely many of these questions (which will obviously be necessary, since $1/3$ has an infinite binary expansion), and the object in which the infinite intersection of the decreasing family of closed intervals with rational endpoints constructed by answering the sequence of questions contains precisely one point is called the real numbers. Hence one ends up with the real numbers as describing locations on the line, while not actually being the line itself.

In fact, the construction of the real numbers also gives you some "lininess" as baggage from the construction: you produce a topology, which tells you about locations being close to one another. This gives the real numbers more "substance" than just being ordered and containing the rationals. One can define topologies on the rationals, but the real numbers' completeness in their topological construction is the key. Completeness forces there to be "too many" real numbers to be covered by arbitrarily small sets. (Obviously countable is too small since the rationals don't work, but the Cantor set shows that one can produce uncountable sets with zero "length".)

One large hole in this so far is what "length" actually is. To do things this way, one is forced to introduce a definition of the length of a rational interval $[p,q]$, which must of course be $q-p$. Since one is not concerned at that point about the interval actually being "full" of points, one can simply introduce this as an axiom of the theory: all of us at some point have owned a ruler and know how they work with integers and small fractions, and it's not too much of a stretch to stipulate that one can have a ruler with as small a rational subdivision as required, without having to resort to infinite subdivision. (Which is another point worth emphasising: without infinite processes, there is no need for the real numbers in toto: one can simply introduce "enough" rationals for the precision one requires, and work modulo this "smallest length".)

This way, one starts with "length" and ends up with "real numbers", rather than trying to go the other way, which is theoretically difficult and mentally taxing and counterintuitive (besides all the Cantorian stuff).

Answer 2

At that level, you can only define length in terms of line segments, which should not present a problem.

To approach the idea of "length" of a point you could use the idea of probability.

You might use an example such as this. Suppose we randomly choose a number $x$ in the open interval $(0,1)$.

1. What is the probability that $x$ will be in the interval $\left(\frac{1}{2},1\right)$?
2. The interval $\left(\frac{1}{3},\frac{2}{3}\right)$?
3. What is the probability that $x$ will exactly equal $\sqrt[3]{\frac{3}{\pi}}$?

You can then relate the idea of "length" to probability.

Answer 3

Perhaps you create a mapping from the length of each segment to the number of segments necessary, namely $n= (1 \ \mathrm{cm})/{\ell}$. Graph this function with $n$ on the vertical axis and $\ell / \mathrm{cm}$ on the horizontal axis. Show them that where $\ell = 0 \ \mathrm{cm}$, which is the “length” of a point, it is implied / we can conclude / for the purpose of completeness, we declare / et cetera that $n=\infty$.

I think this would be a wonderful introduction to limits. I think you could reinforce this claim by proving out or proving that the graph is valid for all other positive values of $n$. I know that this helped me think through calculus when I was learning.

Answer 4

If you have a full period to use, and your students are bright, you could explain the why integers are countably infinite while irrationals are uncountably infinite. Once you'v done that, you can point out how adding up the lengths of a countably infinite number of points does indeed get you zero (0+0+0..=0) but that a line segment doesn't have a countably infinite number of points - it has an uncountably infinite number of points.

That doesn't explain how the points create a line, but it does explain why the main objection - adding zero length to zero length gets you zero - doesn't apply.

Answer 5

First off, consider instead trying to argue that a line can be made up of infinitely many small lines, each of infintessimally short length. This will be closer to the way calculus handles lines, so will serve them well in the future.

If you must use points, one approach you might be able to take is to start with two points, $0$ and $1$. Then draw in the mid point ($\frac{1}{2}$). Then draw in the $2$ midpoints from there ($\frac{1}{4}$ and $\frac{3}{4}$). Then the $4$ midpoints, then $8$, and so forth. In each stage, you draw more and more midpoints. If you do this an infinite number of times, you will fill in the line completely. (I'm pretty sure you just need to do a countably infinite number of steps here, for iteration $N$ of this midpoint filling process fills in $2^N$ points, so by the continuum hypothesis, that process will produce a number of points equal to the cardinality of the real numbers. Stronger mathematicians, please check!).

As a bonus, this process visually fills in the line rather quickly, while the formalism showing that the line is connected at the end of the process can be left for higher mathematics.

Answer 6

Elsewhere in comments, you've suggested you're familiar with topology. In these terms, I think I can describe more precisely what the issue is.

Consider the usual subspace topology on the interval $[0,1]$ of the real line, and also the discrete topology on the same set of points.

The notion of "an infinite collection of points" is really describing the latter topological space. It's only by considering those points in place as describing a subspace of the real line that we get something with line-like qualities.

If we want to consider the points in isolation, we also have to remember the relevant structure (e.g. topology, metric, or whatever) if we want to talk about the points having any line-like qualities.

So that's what's going on — a line isn't made out of points, and it's the extra bit the students are overlooking, such as a metric, that actually makes the set into a line segment one centimeter long.

As to how to explain the difference to students... that's why people are suggesting you ask your question at http://matheducators.stackexchange.com!

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One could try to find a way to explain the difference between countable additivity and uncountable additivity of measures, but I strongly expect that would be missing the point. (also, measures forget almost everything about geometry, so trying to use them to explain how a set of points can be linelike is futile)

Answer 7

You write: "As an undergraduate I assumed it was due to having uncountably many points. When I learned about measure theory, I realized that's not the explanation."

But in fact this is indeed the explanation. Lebesgue measure $\lambda$ is countably additive (in ZFC) so if $\mathbb R$ were countable one would indeed have $\lambda(\mathbb R)=0$. But countable additivity is not generalizable to any hypothetical notion of uncountable additivity. Therefore no paradox of the sort $\lambda(\mathbb R)=0$ arises.

Answer 8

I agree with Mikhail Katz: you were right as an undergraduate when you "assumed it was due to having uncountably many points." Since you say you have learned measure theory, I am not sure why you discarded this (correct) explanation.

To those who are less familiar with differing infinite cardinalities, a simpler explanation might be that you cannot necessarily add up an arbitrary number of sets and have the lengths add. It is true that if you add one line segment to another, the lengths add. But the sets must be special (if asked further: "measurable"), and to add an infinite number of things there must be a technical restriction (if asked further: that there only be countably infinitely many).