Imagine a sphere entirely above an infinite flat plane. From any point on the plane, draw an infinite straight line through the center point of the sphere. The line will intersect the sphere at two points. These points are unique for each point on the flat plane surface. In cartesian geometry, the plane passing through the center of the sphere parallel to the exterior plane will intersect a circle on the sphere which no point on the plane can touch by the above method. Therefore the surface of a sphere has more than twice as many points as an infinite surface.
There is a 1-1 correspondence between, for example, the integers and the rational numbers, even though the integers form a strict subset of the rationals.