- [The CTMU] nowhere relies on naïve set theory, and in fact can be construed as a condemnation of naïve set theory for philosophical purposes.
- As it happens (and not by accident), consistent versions of set theory can be interpreted in SCSPL. The problem is, SCSPL can’t be mapped into any standard version of set theory without omitting essential ingredients, and that’s unacceptable. This is why the CTMU cannot endorse any standard set theory as a foundational language. But does this stop the universe from being a set? Not if it is either perceptible or intelligible in the sense of Cantor’s definition.
- The CTMU does not rely on set-theoretic self-inclusion to effect explanatory closure (as it would have to do if the universe were merely “the largest set”).
- No axiomatic version of set theory is included in Cantor’s definition of "set"; the definition is meaningful without it. Rather, the definition of "set" is itself a "minimal set theory" which can be viewed as a general subtheory in which multiple set theories intersect; it differs from standard set theory mainly in omitting the self-inclusion operation, and the further operation of compensating for that operation.
- The power set is a distributed *aspect* of the universe by virtue of which objects and sets of objects are relationally connected to each other in the assignment and discrimination of attributes (the intensions of sets). Without it, the universe would not be identifiable, even to itself; its own functions could not acquire and distinguish their arguments. In fact, considered as an attributive component of identification taking a set as input and yielding a higher-order relational potential as output, it is reflexive and “inductively idempotent”; the power set is itself a set, and applied to itself, yields another (higher-order) power set, which is again a set, and so on up the ladder.
- Of course, even the perceptual stratum of the universe is not totally perceptible from any local vantage. The universe, its subsets, and the perceptible connections among those subsets can be perceived only out to the cosmic horizon, and even then, our observations fail to resolve most of its smaller subsets (parts, aggregates, power-set constituents). But a distributed logical structure including the power set can still be inferred as an abstract but necessary extension of the perceptual universe which is essential to identification operations including that of perception itself.
- The scientific import is obvious. Where the universe is defined, for scientific purposes, to contain the entire set of past and future observational and experimental data, plus all that may be inferred as requirements of perception, its power set is integral to it as a condition of its perception and scientific analysis, not to mention its intrinsic self-differentiation and coherence. Without its power set, its parts or subsets would be intrinsically indiscernible and indistinguishable, which would of course amount to an oxymoron; “parts” are distinguishable by definition, and therefore constitute a set with the discrete topology construed by relevance (any reference to which naturally invokes the power set) and the indiscrete topology construed by veracity (inclusion-exclusion). Without the power set function and its stratified relational potential, one not only can’t say what the parts and their mutual relationships are, one can’t even say what they’re *not* … and as any parts not relevant to the others are not “parts” as advertised, even referring to them generates contradictions and must therefore be avoided.