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John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
http://www.pitt.edu/~jdnorton

This chapter will analyze a collection of (initially) perplexing paradoxes concern the measure--the length--assigned to curves. They reside in arguments that purport to show that that length of a straight line is both 2 and π at the same time, so that π = 2; and one that purports to show similarly that √2 = 2.

The paradoxes are included here since they concern matters of measure, which has been the topic of preceding chapters. While they are concerned with measure, the solution to the paradoxes does not involve issues specific to measures. Rather the paradoxes arise from an incautious use of infinite limits. Below we will first see the paradoxes and then proceed to their solutions.

# "Proof" that π = 2

The word "proof" is enclosed in so-called scare quotes to flag that it is not a cogent proof at all.

The "proof" starts by considering a straight line interval of length 2 and a serpentine curve that very roughly approximates it. The serpentine curve consists of two semicircles connected together.

Each semicircle has a diameter of unit length, so that the full circle has the circumference π. It follows that the semicircle has a length of π/2, so that the full serpentine curve has a length of π. Call this first curve "C1."

The curve C1 is not a close match to the straight line. We can improve the fit by replacing it by a second curve, C2. It is formed by taking the curve C1 and uniformly contracting it in both height and width to half its size. We then attach a copy of the contracted curve to form the even more serpentine curve C2.

Another way to see it: each semicircle has diameter 1/2, so that the semicircle has length π/4. There are four semicircles, so the total length is 4 x π/4 = π.

The general result for all curves is proved here.
Word source document here.

It is not too hard to see that the curve C2 has the same length, π, as the original curve C1. When a curve is uniformly contracted in both width and height to half its size, its length is also contracted to half its original length. This means that the contracted curve has length π/2. When we extend it with a duplicate of the same length π/2, we end up with a curve of total length π.

We now continue the process by contracting curve C1 to one third its size and extending it with two copies of itself. What results is the curve C3 that is even more serpentine.

The construction continues to give us the infinite family of curves:

C1, C2, C3, C4, C5, ...

Here is curve C9:

This family of curves has two properties:

• The length of each curve is the same, π.
length(C1) = length(C2) = length(C3) = ... = π.
• The curves come closer and closer to the straight line interval,
such that the infinite limit of the curves coincides with the straight line segment.

Since all the curves are same length π, it follows that the straight line interval is π also in length. That is, we have shown that:

π = 2

# "Proof" that √2=2

The "proof" proceeds in much the same way as the "proof" that π = 2. We start with a straight line of length √2. It is approximated by an inverted V.

The two equal-length arms of the inverted V form a right angled triangle with the hypotenuse coinciding with the original straight line of length √2. This is the curve C1. It follows from Pythagoras' theorem that each arm of the inverted V is of length 1. To total length of C1 is 2.

This single inverted V, C1, is a poor approximation to the length. We replace it by uniformly shrinking the inverted V to half its size and duplicating it to form a new curve C2. The two smaller inverted V's are a better approximation to the original length. Each shrunken inverted V has half the length of the original V. Two of them together has the same length. Hence the length of the curve C2 is also 2.

We repeat the procedure by replacing each of these inverted V's in C2 by two inverted V's of half the size to arrive at curve C3. Repeating the replacement again and again, we generate an infinite set of curves C4, C5, C5, ...

As we proceed along this sequence of curve, they come closer and closer to the original length and, in the infinite limit, coincide with it. However the lengths of all the curves are the same as the length of the original inverted V, C1, which is 2:

length(C1) = length(C2) = length(C3) = ... = 2

Thus we conclude that the length of the limiting curve is 2. However by construction the original length was √2. It follows that

√2 = 2

The collection of inverted V's has many sharp kinks. One might wonder if, in this case, something untoward happens because of them. As we take the limit of infinitely many of the curves, we will have infinitely many sharp kinks.

We can set aside these concerns by noting that it is a minor adjustment to introduce a small amount of rounding of the curves to smooth out the kinks. This can be done so that the rounding is always a small portion of the whole curve and thus does not affect the overall result. For example, here is the second curve, C2, with the roundings.

# The Limit Curve

To begin the resolution of the paradoxes, we need to make explicit the delicate step in the argument. In both we first define an infinite sequence of curves:

C1, C2, C3, C4, ...

We then define is a new curve, the "limit curve, C," that results from taking the limit of the curves to infinity. This limit curve is not any one of these finitely numbered curves. Each of them is serpentine or kinked, which means that none of them can coincide with the straight line we are identifying as the limit curve, C.

Rather, this limit curve is constructed from the set of curves by the procedure illustrated in the figure below (for the case of the "proof" of √2 = 2). We identify corresponding points in the set of curves by drawing a vertical line through the family of curves and noting the intersections of the vertical line with the curves.

In the figure, they are P1, P2, P3, ...  These points of intersection cluster together and come arbitrarily close to a point on original straight line. That point, p, is the limit point for this particular set of points of intersection. If we repeat this construction for all the points in the family of curves, we find that the limiting curve, C, is just the original straight line.

# Limits, Well Behaved

What this last construction tells us is which points comprise the limit curve. It does not yet tell us what the length of that curve is. That omission will become a central issue. It is tempting to imagine that the length of the curve is just automatically carried along in the limiting construction with the set of points that comprise the curve.

In ordinary cases, no trouble is caused by this assumption. We proceed assuming that the limit set of points inherits its other properties as expected from the limiting process. These are the cases of "well behaved" limits. Here is a simple example.

Recall: (0, 3/4) denotes the set of real numbers between 0 and 3/4, but excluding 0 and 3/4. Hence "open."

Consider the open sets of real numbers

(0,1/2), (0, 2/3), (0, 3/4), (0,4/5), ...

As we proceed along this infinite sequence of open sets, the sets come arbitrarily close to open set (0,1). However (0,1) is not in the sequence. It is the limit set to which the sequence converges.

Now consider the length assigned by the familiar (Lebesgue) measure to the open sets. The lengths are just 1/2, 3/4, 4/5, 5/6, ... and the limit of that sequence of measures is just 1. That limiting value also is the length of the limit set, 1.

Everything is proceeding as we expect it should. We have formed a limit set, (0,1) and we have formed a limit of the lengths, 1. And that limiting length is the length of the limiting set.

(well behaved) The length of the limit of the sets agrees with the limit of the lengths.

Perhaps the point is so obvious that enough has been said. However--since this nice behavior is what  will fail--it is worth spelling it out in a table:

 set (0,1/2) (0,2/3) (0,3/4) (0,4/5) ... limit set (0,1) length 1/2 2/3 3/4 4/5 ... limit length 1

Good behavior like this is common, but cannot be assured. Bad behavior of the limit arises when the limit set and the limit properties disagree. When this happens, it is often associated with a sense of paradox. We have already seen an example of this bad behavior in the Ross-Littlewood Urn Supertask. Here is a comparable summary of the results

 time -1 -1/2 -1/3 -1/4 ... 0 ball numbers in urn 2 to 10 3 to 20 4 to 30 5 to 40 ... no balls in urn number count 9 18 27 36 ... ∞

The general situation in which this bad behavior arises is that we have an infinite family of systems that produces some limit system. The systems have properties; and limit of those properties fails to match the property of the limit system.

In the case of the Ross-Littlewood urn supertask, the systems consist of the balls in the urn at successive times. The property is the number count of balls in the urn at successive times. The limit  system at time 0 is an empty urn, that is, one with number count zero. The limit of number counts of the systems, however, is infinite.

(badly behaved) The property of the limit system does not match the limit of the properties.

Applied to the particular case of the curves of this chapter it is:

(badly behaved)  The length of the limit of the sets does NOT agree with the limit of the lengths.

# Badly Behaved Limits of Curves

The paradoxes of these dubious "proofs" arise directly from employing badly behaved limits. Here is the first "proof" that π=2.

 curve C1 C2 C3 C4 ... limit curve is a straight line of length 2 length π π π π ... π

The limit of the lengths is π. This limit property does not match the length, 2, of the limit system. The false assumption that generates the paradox is that the limiting procedure is well behaved. If we make this false assumption, we infer in error that that the length of the limit curve is π.

Is the limit of the lengths really π? It is and it is an especially simple case of such a limit. A more familiar example is the sequence

1/2, 2/3, 3/4, 5/6, ... → 1

Its limit is 1 since if we pass far enough along the sequence, we come arbitrarily close to 1. Now consider

π, π, π, π, π, π, ... → π

The same result applies. If we pass far enough along the sequence we come arbitrarily close to π. However now the condition is satisfied trivially since at any stage of the  sequence we are already exactly at π.

Here is the second "proof" that 2 = √2.

 curve C1 C2 C3 C4 ... limit curve is straight line of length √2 length 2 2 2 2 ... 2

The limit of the lengths is 2. This limit property does not match the length, √2, of the limit system. As above, the false assumption that generates the paradox is that the limiting procedure is well behaved.

Repeating the analysis using the calculus does not seem to provide any further illumination. Details here. Word doc source here.

# An Infinite Length in the Limit

The paradoxes depend on the length of the limit curve differing from the limit of the lengths of the curves that generate the limit curve. Once we see that this separation of values is possible, we can ask just how great the separation can be. Here is a case in which we use an infinite family of curves to approximate a finite length, but the limit of the lengths of the curves in the family is infinite.

The example arises as a modification of the "proof" that 2 = √2. In the original "proof" we uniformly reduced each inverted V by halving both its height and its width. What resulted was a smaller inverted V of half the original length. To proceed from one curve in the family to the next, each inverted V was replaced by a pair of these smaller inverted V's.

In the present variation, the shrinking of the inverted V's is carried out slightly differently. Each inverted V is halved in height, but the width is reduced by a factor of 4. To proceed from one curve to the next, each inverted V is replace by 4 such reduced inverted V's, chained together. (In the figure below, the dashed lines mark full height, half height, quarter height and eighth height.)

Under this new scheme, each inverted V is not replaced by a reduced pair of the same length, but by four reduced V's that have a greater length. One might imagine that the four reduced V's have twice the length of the original V. That is almost right. Each set of four V's is almost twice as long as the original; and, as we proceed along the curves, they approach arbitrarily closely to double the length of the single inverted V in the preceding curve that they replace.

The calculation of the precise lengths requires only the repeated use of Pythagoras' theorem. The simplest way to recover the growth in length is to compute each directly. They are calculated in a spreadsheet here.

The outcome is that, as we proceed along the curves, their lengths continue to grow. With each step to a new curve, once we proceed past the first few curves, the curve lengths almost exactly double. It follows that the limit of the lengths of the curves is infinite. Here is the behavior:

 curve C1 C2 C3 C4 C5 C6 C7 ... limit curve is a straight line of length √2 length 2 3.16 5.83 11.40 22.67 45.28 90.52 ... ∞ curve length increases by factor 1.58 1.84 1.96 1.99 2.00 2.00

if we treat this construction as a "proof" it gives us the result that

∞ = √2

# Finite Length but No Limit Curve

This chapter now concludes by taking the paradoxical construction to the opposite extreme. In all the examples above, the limit of the curves is well-behaved. It is the limit of the lengths that is not well behaved, in the sense that its limit differs from the length of the limit curve. Here we shall see a case of the limit of the lengths behaving well. However the family of curves will be badly behaved in the sense that there is no limit curve at all.

The construction begins with a straight line of unit length. As before it will be approximated by an inverted V. The inverted V, shown below, has length 1, the same as the length of the straight line. However it extends from the half-way point in the straight line to its end. The first half of the straight line is left uncovered.

It will be important below that the set of points forming this inverted V is an open set. In the figure below, the small open circles indicate where the inverted V would intersect the straight line. These two points of intersection are NOT included in the inverted V.

To proceed to the next curve, we take this first inverted V and uniformly reduce it to half its height and width. The reduced inverted V has half the original length, that is, 1/2. To restore the full length of 1, a copy of the reduced inverted V is placed over the first half of the straight line, leaving two quarter length gaps as shown below.

We continue to create a third curve in the same way by replacing each inverted V by two reduced inverted V's of the same total length. The process continues, creating the fourth, fifth, ... curves in the family.

The result is an infinite sequence of curves, the first few of which are shown above. While each of the curves consists of many disjoint parts, when they are summed, the total length of each curve is still 1. It follows that the limit of the lengths of the curves is 1. It agrees with the unit length of the original straight line.

While the limit of the lengths is well behaved, the limit of the curves is not. There is no well-defined curve that is the limit of the curves in this family of disjointed curves. The difficulty is a failure of convergence. It is analogous to the failure of the infinite, alternating sequence

0, 1, 0, 1, 0, 1, ...

to have any definite limiting value. As we proceed along this infinite sequence, we do not come arbitrarily close to a limiting value. We just find terms that alternate endless between 0 and 1.

Something similar happens when we try to form a limit curve here. It is similar, but more complicated. We have seen above the procedure for identifying the limit curve. We draw a vertical line through the curves, identify the points of intersection and then trace down the vertical line to find the limit point of these points of intersection.

To simplify the analysis, the figure below suppresses the vertical dimension for each curve. It displays as a shaded bar the points in the horizontal direction occupied by each curve. For ease of identification, positions in the horizontal direction are indexed by a coordinate x that extends in value from 0 to 1.

The first curve, C1, extends from x=1/2 to x=1. That is, the points occupied form the open set written as (1/2, 1), where the openness of the set means that neither x=1/2 nor x=1 are included in the set.

Similarly, C2 corresponds to the set of occupied points (1/4,1/2) (3/4, 1); that is, it is the union of the two open sets (1/4,1/2) and (3/4, 1).

C3 corresponds to the set of occupied points (1/8,1/4) (3/8, 1/2) (5/8, 3/4) (7/8, 1)

and so on for the remaining curves.

We can now ask which points will be found in the limit set, constructed as above? The full analysis is somewhat taxing and is provided in a document here. Word source document here.

The overall result comes in two parts. First consider points that have x coordinates that are multiples of 1/2, 1/4, 1/8, ... They form the countably infinite set

{1, 1/2, 3/4, 1/4, 7/8, 5/8, 3/8, 1/8, ...}

All of these point are located at the open ends of the intervals forming the curves. Since those end points are not included in the open sets, it follows, as shown in the figure below, that none of them are in the set of points that would form the limit curve.

If the limit process were the same for all the remaining points, then the limit set would be well defined. But it would be the empty set.

The second part consists of all the points not in the above set. They form the bulk of the points that we might find in the limit set. These remaining points form a set of continuum size, which is infinitely greater than the countable infinity of points included in the first part. The points of this greatest bulk elude inclusion or exclusion from what would be the limit set, leaving that limit set ill-defined.

A simple example is x=1/3. We find that 1/3 alternates indefinitely in being a member of the sets associated with the curves and in not being a member of the sets, as we proceed through the family. We have

Recall "∈" means "is a member or element of the set"; and "∉" means "is not a member or element of the set."

1/3 ∉ C1, 1/3 ∈ C2, 1/3 ∉ C3, 1/3 ∈ C4, ...

These alternations mean that we get no definite answer to the question: "Is x=1/3 included in the limit set of points comprising the limit curve?" The answers corresponding to the alternations above are:

no, yes, no, yes, no, yes, ...

That means that there is no definite fact of whether x=1/3 is included in the limit set.

These alternations of "no-yes" arise in the simplest way for the case of x=1/3. All the other points considered in this part exhibit more complicated variations of alternations among "no" and "yes." They all do it in such a way that for all of them no definite answer can be given to the question of inclusion in the limit set.

It follows that there is no limit set and thus no limit curve for the family of curves in this example.

November 23, 2021