Another way to think of dimension is as the degrees of freedom available within the space. Our choice of directions must be orthogonal , however. Motion in a particular direction cannot be expressed as the finite combination of infinitesimal motions in the other directions. It should not be possible to move up by the finite combination of extremely small left and forward motions. If it is, then we have over-described the dimension of the space.
Any vertical motion is the result of moving in the other two directions.
Under these constraints, a countable set of points is now zero-dimensional as we have zero degrees of freedom. It is not possible to move through such a space from one point to another without leaving the space. This would make the space one-dimensional, which feels more natural. This does not feel natural. So far, my commonsense definitions are not agreeing with my commonsense answers.