If R_s, our universe of discourse, is the multivalued reference space (Julian Barbour’s “Platonia”) and M_s a formalized language over R_s called Musculpt (music-sculpture), then we wish to ascertain the possibility, within the construct of a metatheory over M_s and R_s , of making precise the concept of validity of a semantic assertion of M_s in R_s. Is there a codifiable axiom system, A, such that the set of statements derivable from A by the rules of Post’s “m-valued truth systems”, _µT_m, coincides with the set of valid statements over R_s?

We take seven undefined terms -- number, value, equal, zero, infinity, immediate successor to, hole -- and state five axioms.

The Multivalued Reference Space axiOOOMMitized
(from the point of view of 2-valued logic):

• A number is a value which is not in all cases equal to itself.
• Zero and infinity are both numbers.
• The immediate successor to a number is not necessarily a number (it could be a hole, which is not zero or infinity).
• One value of that immediate successor to a number, which is a number, is always zero or infinity, or both.
• Any property belonging to any number belongs to some value of every number.

The set, S, of operational and functional symbols -- constant signs; numerical, sentential, and predicate variables -- is denumerable, but not finite, because M_s (by virtue of _µT_m ) is a formalized language with a class of mⁿ configurations Each added configuration invokes additional operational and functional symbols by virtue of the changed value (identity status) signified by the audiovisual phonemes, morphemes, and functions of the M_s semiotic.

The consistency and completeness of a 2-valued calculus can be absolutely demonstrated only with reference to a 3-valued logic; that of a 3-valued calculus, only with reference to a 4-valued logic; et cetera. What happens when the order of value of the logic is À₀ and À₁ (the numbering of orders of logical-value includes more than merely the natural numbers)? Two distinct varieties of self-reentry occur. (Note that ½= means follows from the mathematical entity preceding it.) À₀½= absolute transparency of opposites (A is absolutely not-A). And À₁½= self-production of identity: hyparxis. The various properties of self-organization -- which is more complex than self-production -- follow from higher orders of infinity. Autopoiesis is an essential feature of the Cantor universe.

---