Set of all sets
In the CTMU, the set of all sets is interpreted as reality, the entire real universe. It is the largest set, being both all-inclusive and self-inclusive.
In naïve set theory, the set of all sets leads to paradoxes. For example, since every set has a powerset which is larger than the set itself, the set of all sets is smaller than its powerset. But then the set of all sets is not the largest set, a contradiction. To escape these paradoxes, mathematicians have adopted a set of axioms (ZFC) according to which the set of all sets does not exist.
The CTMU takes a different approach, retaining the set of all sets by defining an extension of set theory incorporating a dual notion of containment.
[E]very set, even the largest one, has a powerset which contains it, and that which contains it must be larger (a contradiction). The obvious solution: define an extension of set theory incorporating two senses of "containment" which work together in such a way that the largest set can be defined as "containing" its powerset in one sense while being contained by its powerset in the other.
The two types of containment are topological and descriptive. Informally, topological containment means containing something like a cupboard contains clothes. Descriptive containment means containing something the way that clothes define what the cupboard IS in the first place. Were not every aspect of the cupboard made specifically for the clothes that it was to contain, there would be no cupboard as a concept. In the same way the universe "holds" all the things inside it, but the things inside the universe define what it actually is. (The "dual" relationship between these two kinds of containment is covered at TD duality.)
The name of the extension of set theory incorporating this dual notion of containment is SCSPL. Thus, in the CTMU, the set of all sets is not just a set, but an SCSPL (Self-Configuring Self-Processing Language) corresponding to reality itself, and elements of reality are not just set-theoretic objects, but syntactic operators.
As an SCSPL, reality evolves through a two stage process where one stage is descriptive and the other topological. In stage 1 all of the items in the universe reconsider their previous state, communicate and describe the rules and definitions for a future universe(the descriptive phase, where time operates). Stage 2 involves the physical reality coming into being where objects have boundaries that distinguish them from other objects and all of these objects are "contained" within space (the topological phase, time has become space). The name of the process given to the universe constantly topologically containing itself descriptively containing itself topologically and so on, is conspansion.
- Langan, Christopher M. (1999). "Introduction to the CTMU". Ubiquity Vol. 1, No. 1.