How compelling is this argument for multiple degrees of infinity?

Question

A while back I heard an argument supporting multiple degrees of infinity that I found interesting. It goes as follows:

1. Given a circle, draw every possible radius (hypothetically). Presumably, there should be an infinite number of line segments, with a 1-to-1 relationship between their endpoints and points on the circle.
2. Draw a second circle around the first (so that they're concentric).
3. Extend the radii you've drawn outward so they intersect the points along the outer circle.

At this point, thanks to our good friend geometry, there should be some space (albeit infinitely small) between the endpoints of the line segments along the outer circle, implying that the number of points along the outer circle is greater than along the inner circle.

That seemed pretty convincing to me, but I'm starting to wonder if geometry stops being our good friend once infinity is involved.

I mean, if you had both circles and tried drawing every radius for the outer circle first, it's not like the radii would overlap along the inner circle, right? By definition, those lines won't intersect until they reach the center of both circles, so it seems that for any point along the outer circle, there exists a radius mapping it to a point along the inner circle. That seems to strongly imply this argument is flawed.

However, the argument still makes sense to me, so I don't know how to resolve this contradiction.

Is the main argument relying on a flawed assumption (and if so, that the geometry it uses still applies when dealing with infinity), is my counter-argument flawed, or are both flawed and I just have no idea what I'm talking about?

Answer 1

It's not geometry that stops being your friend so much as your finite-case-based intuition. You mention gaps when you consider the longer circle, but I think you're imagining gaps from finitely many radii.

If you draw a radius for every real number angle between 0 and 360 degrees (and include one more for 0 or 360), then you've covered every angle, regardless of the size of the circles you draw. If there were a missing radius, it would have to have some missing angle, but to cover all the points of the small circle, you need to cover every angle.

If you're comfortable with trigonometry, every point in the whole plane except for the origin has a counterclockwise angle from the positive half of the $x$-axis given by a function sometimes known as atan2, based on, but a bit different from, $\arctan(y/x)$.

Answer 2

Your statement that there should be some space between points on the outer circle is not correct. Each circle has $\mathfrak c=2^{\aleph_0}$ points on it and your construction establishes a bijection between points on the inner circle and points on the outer circle.

Answer 3

First of all there are as many points on the outer circle as there are on the inner circle (both sets of points have the same cardinality as the set of real numbers, as do most geometric figures)

Aside: When comparing infinities you have be careful. Even if an infinite set strictly contains another infinite set, they can still have the same "size" (i.e. cardinality).

Back to your question, the flaws lie in your main argument (your counter argument is correct):

(1) There is no space between the points in the outer circle

(2) Even if there was, it still wouldn't be sufficient to show that it has more points.