number of points on two circles

Question

(sorry I don't know how to add pictures)

Two friends argue if larger circles have more points than smaller circles

Friend number 1 (a well known argument)
Say the circles are concentric. you cannot draw a line from the centre that cuts the bigger circle while that doesn't cut the inner circle so they have the same amounts of points.

Friend number 2 ( a dissenting voice)
Ok lets take concentric circles, he draws a line to the centre ( say along the x axis)
adds another line parallel to this line it cuts both circles 2 times)
adds another (cuts both circles 2 times) and so on
two parallel lines are only tangent to the smaller circle while still cuting the bigger at two points. and some paralel lines only cut the larger cirle.

Therefore all points on the smaller circle are related to some point on the bigger one, but some points on the bigger one are not related to point on the smaller one.
So the larger circle has more points.

Which friend is right or how do you convince them that they are both right?

Answer 1

Friend 1's argument does indeed show that the circles have the same number of points, as each ray from the center uniquely determines a point on the circle, and each point on the circle uniquely determines a ray from the center.

Friend 2's argument (if made carefully) shows that the bigger circle has at least as many points as the smaller circle, but does not show that the smaller circle has strictly fewer points.

It all comes down to the notion of cardinality of infinite sets, where things can get pretty counterintuitive. If you want to give friend 2 an example to show why that argument does not show strict size difference, consider the set of integers $\Bbb Z.$ Obviously the same size as itself, yes? Ah, but now consider the function $f:\Bbb Z\to\Bbb Z$ given by $f(x)=2x$. This function maps $\Bbb Z$ into $\Bbb Z$, but misses infinitely many integers! That does not mean, however, that $\Bbb Z$ is strictly larger than itself. It is simply a peculiar quirk of (many) infinite sets that they can be put into one-to-one correspondence with subsets of themselves. Thus, showing that the smaller circle is in one-to-one correspondence with only a part of the larger circle isn't enough to show that the larger circle has more points--though such an argument would work for finite sets.

Answer 2

I could be mistaken here, but the main conflict seems to be in compromising between the geometrical and topological perspectives.

Geometrically one could say that the arc length of a curve is a measure of how many points it has, so clearly the larger circle has more points.

However from a topological perspective, the cardinality of both sets of points are the same in that any circle can be mapped to the real line via a stereographic projection.

So in a sense, both answers are right!

Answer 3

What does "more points" mean here?

I'll try and argue that both friends come as correct and they both friends have different notions of what "more points" means.

The friend who says that circle "a" (the smaller one) has the same size as circle A, might define an object x to have the same size an object X if and only if someone can or has written a bijection between the points of x and the points of X.

The friend who says that circle "a" is smaller than circle A, might write the following definition:

Part 1: Object x to has the same size as object X if and only if when we make a random selection of a point from [po(x) U po(X)], where [po(x) U po(X)] indicates the union of the sets of points of x and the sets of points of X, the probability that we've selected a point from x equals the probability that we've selected a point from X, or prob(x)=prob(X).

Part 2: Object x comes as smaller than object X if and only if when we make a random selection from [po(x) U po(X)], it comes as more likely that we'll select a point from object X than object x, or prob(x) < prob(X).

Part 3: Object x comes as larger than object X if and only if when we make a random selection from [po(x) U po(X)], it comes as more likely that we'll select a point from object x than object X, or prob(X) > prob(x).

Friend two's argument gives us a procedure as to how to observe that circle "a" has a smaller area than that of circle A (that much, I do believe, all set theorists would grant). So, if a machine not trying to hit either circle threw a dart at the circles standing next to each other, it'd come as more likely that the dart hit circle A than that it hit circle "a" (the possibility that it would miss either of the circles comes as irrelevant). It consequently follows that a random selection of a point from

[po(a) U po(A)] will more likely give us a point of A than a point of "a". So, "a" has a smaller size than A, according to friend two's definition of what size means.

Answer 4

Friend 2 is right in his conclusion but I don't like his method.

What is the flaw in the argument of friend 1? Consider two concentric circles which have radii of 2$\pi$r and 2$\pi$R respectively, with R > r. Let us define a point on a circle to be a segment of arc length dx. Let us draw a line from the center of the circles which then cuts both along the positive y axis (0,1) and then rotate this line clockwise by d$\theta$. Indeed in both circles this rotation has covered an angle of d$\theta$; however, in the large circle it covered an arc length of Rd$\theta$ whereas it only covered rd$\theta$ in the small circle. Recall that the definition of a point involves a 1D unit of length dx. Thus this rotation covered $\mathrm{\frac{R d\theta}{dx}}$ points in the big circle whereas it only covered $\mathrm{\frac{r d\theta}{dx}}$ points in the small circle. Thus, the ratio of number of points between the large and small circles is R:r, which is just the ratio of the circumference. The mistake in the logic of friend 1 is simply that he used a different definition for what constitutes a point for the big circle than for the small circle. If you compare a quantity (number of points) between two objects and change the definition of the quantity from object to object, then the conclusions made are likely to be meaningless.

That being said, many people will use the defective argument that the aforementioned logic is only true until we reach the limit of dx$\rightarrow$0, where we define a point to have infinitesimal width . In which case, the number of points in a circle reaches the corresponding limit "number of points"$\rightarrow \infty$. Then, since $\infty \times n = \infty$ both circles have the same number of points.

The flaw in this logic is that it incorrectly uses the concept of infinity. The statement $\infty \times n = \infty$ can be rephrased to mean "Lack of precise information" $\times$ "a real number" = "Still a lack of information". To use such a formula to justify a conclusion is unfortunately gibberish. It does not matter whether we lack precise information about the point size dx, all that matters is that we know that dx is the same value when applied to the small circle as when applied to the large circle. So what can we say? If indeed dx is infinitesimal (dx$\rightarrow$0) then we do know that there are infinite points on the big circle and the small circle, in the sense that we lack knowledge of the exact number for both circles. But to say, a lack of knowledge = a different lack of knowledge is obviously wrong. The concept of their being different sized infinities is nothing new, so the idea of two different infinities being in an exact ratio should not be surprising. The property of infinity $\infty \times n = \infty$ is only correct when one uses a definition of infinity which breaks all the rules of algebra and is only helpful as a concept. I presume that most of the confusion here lies in the fact that there are lots of types of infinity, so the symbol $\infty$ refers to a useful concept (a number which goes on forever which we lack precise information on), which is usually different to the other infinities we encounter that we actually do have information on.