The Set of all Sets is a Member of Itself.

1. A set exists. This set is equivalent to P=P and therefore is circular. As circular it is intrinsically empty of meaning in and of itself.

An example of this set can be presented as a line. A line is composed of further lines thus is a set (ie a set of lines). Each line simultaneously represents a loop considering both the beginning point and end point are the same.

2. The set as intrinsically empty inverts into a new set. This new set is Q=Q. Q=Q is a variation of P=P as ((P=P)=(P=P)). As circular, through P=P, Q=Q is intrinsically empty of meaning in and of itself unless it Inverts into a new set and this set Inverts into a new set ad infinitum.

An example of this set would be a single line inverting to two lines. One line exists composed of 2 1/2 lines. Each line is 1 line as 1/2 of the original. As the set of line converts to further lines so does the original line progress from 2 1/2 lines to 3 1/3 lines to 4 1/4 lines, etc. Each line, as a fraction of the original yet a singular line in itself, inverts to further lines as well so that fractions contain fractions and multiples contain multiples.

3. P=P contains Q=Q. Q=Q is a variation of P=P, thus for P=P to contain Q=Q, P=P must contain itself as ((P=P)=((P=P)).

An example of this would be 1 line containing 2 1/2 lines. Each 1/2 is a singular line in itself thus equal to the original in the respect both the original line and the variation are equal to 1 line on their own terms. 1 contains 2 thus 2 as a variation of 1 contains 1. Each subset, as a variation of the original, contains the original as both a fraction and fractal. The set contains itself through fractions and fractals which are multiples of the original set. A subset therefore is a recursion of the original set and as such the set contains itself.

4. The set as containing itself is equivalent to an empty loop containing an empty loop. This loop within a loop is synonymous to a line within a line. This is considering the line contains the same beginning and end points as the 0d point itself. Each line, or loop, within a line, or loop, is the same set repeating itself through further subsets. This summation of lines, or loops, as both the set and the set within the sets necessitates an inherent emptiness of all sets.

This can be pictured under a series of rings within rings. Each ring within a ring necessitates both the outer ring as intrinsically empty, through the rings it contains as being empty, as well as being full of rings. The outer ring is thus simultaneously empty and full of rings; therefore what we understand of the set is a dynamic entity where it is a means of inverting one set into another through a series of fractions and fractals. The set is an observation of recurssion, as repetition, and isomorphism, as the inversion from one symmetrical state into another.