(related to Definition: 1.01: Point)
A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor a line; hence it has no dimensions — that is, it has neither length, breadth, nor thickness.
The modern definition of a point is a set-theoretic one. It is itself a set, which is an element of another set. Because a point contains no other elements and because of the uniqueness of the empty set, for its own, it can be identified with the empty set. Of course, there are "different" points in the plane, therefore they cannot be unique in the context of a set, the elements of which they are, but from the set-theoretical point of view this is not problematic. According to the axiom of extensionality, a set is only determined by the elements it contains, not by the sets, elements of which it is. Therefore, there are "different" points in a plane only in the sense that we assign to the point different coordinates.
