"Real line" is used to mean real axis ["Real line" is used to mean real axis, i.e., a line with a fixed scale so that every real number corresponds to a unique point on the line. (http://mathworld.wolfram.com/RealLine.html) There are two basic states that stand in the basis of the real-line, which are: a) = (self identity). b) < or > (no self identity). Let x be a real number. Any real number, which is not x cannot be but < or > than x. The difference between x and not_x, defines a collection of infinitely many unique real numbers. The magnitude of this collection can be the same in any sub collection of it, which means that we have a structure of a fractal to the collection of the real numbers. In short, each real number exists in at least two states: a) As a member of R (local state). b) As an operator that defines the fractal level of R (a global operator on R). Any fractal has two basic properties, absolute and relative. The absolute property: Can be defined in any arbitrary level of the fractal, where within the level each real number has its unique "place" on the "real-line". The relative property: Any “sub R collection” in this case is actually R collection scaled by some R member as its global operator, and this is exactly the reason why some "sub R collections" can have the same magnitude as R collection. We can understand it better by this picture: http://www.geocities.com/complementarytheory/Real-Line.pdf In short, R collection has fractal properties. What do you think?