I thought I carefully checked the answer, because I thought the answer was yes. The sites I looked at seemed to say no, so I went with that. Maybe they were making the distinction that @Chronos made and I misunderstood.
My answer to @Kent_Clark was correct in the difference between the rationals and the reals and I should have stopped there. And stop responding to math threads entirely, great as the temptation is.
The correct answer to the OP is the one already given: The set of lines through the center of a circle and the set of lines through the center of a sphere have the same cardinality, meaning that, with a little work, you can establish a one-to-one correspondence (a bijection) between them. That is, they are the same “size” of infinity. They also have the same cardinality as the real numbers \mathbb R, the plane \mathbb{R}^2, any n-dimensional space \mathbb R^n, and the surface of a sphere of any dimension (sorry @Exapno_Mapcase, there is no difference in cardinality between the surfaces of the various spheres).
That having been answered, let me confuse things by giving a way in which the sets of lines through the center of a circle and the center of a sphere have different sizes. These sets have “shapes,” or topologies. A line through the center of a circle can be identified with the two antipodal points through which it crosses the circle, so we can think of the set of such lines as the circle itself with opposite points identified, which just gives you another circle. Similarly, a line through the center of a sphere can be identified with the two antipodal points where it crosses the sphere, and the set of such lines can be thought of as the sphere with opposite points identified. That doesn’t give you another sphere, it gives what’s called a projective plane. (We know it’s not a sphere because it contains, for example, a copy of the Möbius strip.)
The circle and the projective plane are examples of manifolds, shapes that look locally like pieces of a line, a plane, 3-space, or n-dimensional space \mathbb R^n in general. The surface of a sphere is another example of a 2-dimensional manifold: If you’re standing on the surface and can see only a short distance, it looks flat to you, like a piece of a plane, you don’t see the whole thing curve over on itself.
But manifolds have well-defined dimension: A circle is 1-dimensional and the projective plane is 2-dimensional, and it turns out that you cannot define a continuous bijection between them because of that. It’s in that sense that you might want to say that the two sets have different sizes. They have the same cardinality, but not the same dimension.
Moral: When you’re comparing sizes of things, you have to be very careful to say what you mean by “size.”
Provided, of course, one means integer dimensions, rather than, say, transfinite dimensions.
Which is of course part of why I included the qualifier “usually”, in my first post:
Well, you can still have a countably infinite number of dimensions. If the dimensions are a, b, c, ..., and the digit number as the subscript, then you can encode the number as 0.{a_1}{a_2}{b_1}{a_3}{b_2}{c_1}{a_4}{b_3}{c_2}{d_1}..., where you have groups of 1, 2, 3, … digits and the first digit of dimension N occurs in the Nth place of group N.
You start to wonder whether the cardinality (a measure used to compare sets) really tells you much about how “big” the space is here. For example, consider an infinite-dimensional sphere or projective space. Then it will no longer be compact (I think? @Topologist ?)
Point is, the topology and geometry of these configuration spaces seem to be of some theoretical and applied interest, but merely counting points does not capture any of that.
Yeah, sometimes cardinality isn’t really what we care about. There are other notions of size, like measure theory, which is sort of a generalization of ideas like length, area, and volume.
This is incorrect. The cardinality of the surface of a sphere of any finite dimension is the same as the cardinality of the real line, of the whole space of any finite (or even countable) dimension or actually of any dimension so long as the dimension is not larger than c, the cardinality of the real line (aka, the continuum). Even c^{aleph_0} the set of all integer indexed sequences of natural is no larger than c. c^c, however, the set of all functions from c to c, is definitely larger. On the other hand the set of all continuous functions is not larger, since two continuous functions that agree on all rational numbers are equal.
Hmm… How about \mathbb{Q}^3 vs. \mathbb{R}^2? Which of those is “larger”? Can you sensibly define topological dimension on a space of rational-coordinate points?
That’s correct. For an infinite-dimensional sphere, a covering by open hemispheres won’t have a finite subcover, hence it isn’t compact. (That’s off the top of my head, so if someone comes along and points out I’ve missed some technicality, I won’t be surprised.)
I think one problem here is that the rational numbers, topologically, do not look like the real numbers: they are totally disconnected and their “topological dimension” will be zero.
However, that is not the only mathematical tool at our disposal, so if you are just going to mess with the underlying field you have to eventually come around to an algebraic point of view where it does not really matter and you can regard \mathbb{Q}^3 and \mathbb{R}^2 as algebraic varieties, where the first one is three-dimensional and the second one two-dimensional.