The only thing I know about cardinality of infinities is from an Automata Theory course, trying to talk about which languages can be enumerated by a TM. But I don’t remember why we cared…

Anyway, for the warm-up #1:

My understanding is that circles don’t have a width, so two circles can be arbitrarily close together without touching and that’s ok. So they can nest inside each other, with the “top” of each circle being at the “next” real number down from the outer one, and being just a tiny bit smaller in radius to compensate so it doesn’t run in to the outer one on the bottom. Then we have a number of circles equal to the number of points on the line from the center of the plane out to infinity, and this just the same as saying that a number line has uncountably many real numbers via the Cantor diagonal thing.

For number two (EDIT: disregard, I had the wrong definition of a “disk” as discussed below):

My understanding is that a disk is a circle with a finite width. If that’s so, then it reduces to the same number line question except we *can* enumerate all of the possible disks because the “next” disk’s top is defined by the width. So this is the same as saying that a number line as countably many natural numbers.

For the actual puzzle:

Does a figure 8 have a width? If yes, then it’s trivially the same as a disk isn’t it? If no, then the issue is that there’s spacing implied by the nesting restriction (that the inner 8 has to live entirely inside the top or bottom half of its parent). Put another way, there’s a restriction on the size of the nested 8 relative to the its parent, whereas an inner circle can have *any* radius smaller than its parent. However, I think this doesn’t matter because if we look only at the tops, there can be a new top of an 8 at every real number. The bottoms of those 8s are spaced out, but the tops are uncountably infinite just like the real numbers.

That said, we’re not really filling the plane with 8s by only making use of the top. The bottom of each 8 is empty and you can maybe nest more 8s inside the tops via rotation? However, I’m not familiar enough with the set theory here to know if there’s a relevant next cardinality we can get to here above uncountably infinite. Adding a finite multiple obviously does nothing for us, so I don’t see how the bottoms will be helpful to fill in because that only gives each parent two children instead of one. Even with some rotation tricks, I think we would have to achieve some non-finite amount of nesting at each level to say that there’s “more” of the 8s than real numbers? Otherwise each real maps to a finite set which doesn’t seem different to me.