We are getting better insight into how the multivalued reference space has to be constructed on Hilbert space under m-valued logics. Musculpt and the hierarchy of c-s, c'-s, and c''-s are right at the heart of it. If these insights are pulled out of their m-logically-valued context and put into 2-valued syllogistic logical march via written notation, they will have lost all their intrinsic meaning. On the contrary, cognition has to be pulled out of march in logical syllogism and let fall into Musculpt. Absent Musculpt as mathematical notation, circular presentation is the only real approach, because engagement with it forces the visualization pre-requisite to conscious emergence of always-there subliminal Musculpt (which conventions of written notation deny). Our tornado-genesis-related speculation that interval spread in the hierarchy of c-s, c'-s, and c''-s is in natural log distribution clearly relates to the N/logN distribution of primes, made more precise by Euler's zeta function relative to only the real numbers. Riemann generalized Euler's function to the imaginary numbers and identified a critical band within which all the primes must fall. That they all fall on a line within this band is the famous Riemann Hypothesis. In order to encompass the hierarchy of c-s, c'-s, and c''-s, the Riemann zeta function will have to be generalized to hypercomplex numbers (à la Charles Musés) and the critical band will have to be mapped on multiple sheets relative to m-valued logics. (Distribution of c-s represented by Euler's zeta function relative to the real numbers; distribution of c'-s represented by Riemann's zeta function relative to complex numbers; distribution of c''-s represented by a zeta function relative to hypercomplex numbers.) In this manner, issue of trivial versus non-trivial solutions (sets of zeros) will disappear. What will happen is this: on each prime (arrayed on Riemann's line) on the m-logically-valued reference sheet will be stacked other primes from the multiplicity of single-logically-valued sheets composing the Riemann surface map of Everett’s universal wave-function. (In this approach, Cantorian fractal spacetime relates to the stack of single-valued sheets, which, in turn, relate to Sakharov’s collapse/anti-collapse multi-sheet model of the universe.) On each of the multiplicity of decomposed single-valued sheets, Riemann’s line will be located differently within the critical band than it is located on the m-logically-valued reference sheet, such that, when the complete superposition of numbered sheets is considered, the line will have spread across the whole critical band on the reference sheet (as a result of expanding consideration from single-valued logic to logics of m-values). Because the hypercomplex zeta function would represent distribution of limiting velocities, accelerations, and time rates of change of acceleration, the waveform configured by the distribution step function would be an idealized chronotopological invariant characteristic of the genus (connectivity) of that universal covering surface which is the reference state of a perfectly efficient autopoietic process in optimum self-correlation (which is anything but a catastrophe! to all those not identified with the ego-complex). A supradense m-logically-valued Hilbertian reference space constructed in this fashion has nonlocality of embedded objects as a fundamental property. Locality is a decomposition issue involving cycles of self-reentry (or, alternatively stated, of cosmological self-forgetting, amnesis -- while recomposing the m-logically-valued reference space is Plato’s anamnesis). Lesser levels in efficiency of autopoiesis have chronotopological invariants based on proper subsets of the primes, each with their characteristic step functions and waveforms. (Though these subsets can be put into one-to-one correspondence with the set of all primes, their distribution patterns vary.)
Gödel used prime numbers to encode each element of a logical statement. He defined a specific prime number as corresponding to the “equals sign”, for instance. And another for “plus”, and another for “minus”, and so on. The product obtained by multiplying together all such prime number factors (which, taken together, encode the given logical proposition) is called a Gödel number. So, all possible statements have their equivalent Gödel numbers. Any real number can be factored uniquely into a product of prime numbers. This Gödel methodology for proving his famous theorem is, of course, a very sophisticated form of gematria, sharing properties with Kabbalah -- the portion of gematria missing from Gödel's treatment being the point-set topology associated in Kabbalistic thought with each Hebrew, Greek, or Sanskrit letter. In the present context, consider Stan Tenen’s discoveries, and consider them relative to operator-time. (Operator-time would be characterized by the hypercomplex zeta function, which we believe our canonical equation for harmonic temperature oscillation of p-electron parcels of superconductant DNA can be understood as -- via Julia’s work on primes and critical Hagedorn temperatures.) Tenen’s topologically configured strips (gematric equivalents to the first sentence of Genesis in Hebrew) are Gödel-encoded segments of Riemann’s critical band (as it meanders the multi-sheeted Riemann surface) corresponding to invariants of classes of temporal operation on the m-logically-valued reference space. So, were the Gödel-numbered point-set topologies of Gödel-encoded m-logically-valued propositions to be Riemann surface mapped, every point on the multiple single-valued sheets would have a number associated with it -- some prime, some non-prime. Logical propositions of m-1 and less values would be equivalent to lattices of prime-number points, the values of which are multiplied together (such multiplication being very much like neuronal firing pattern sequences mapped on Szentagothai’s multi-sheeted model of the cerebral cortex). The Gödel numbers resulting from this multiplication would be located on points on the m-logically-valued reference space (the most densely point-packed Riemann surface sheet). The fully-Gödel-numbered m-logically-valued reference space would, thus, contain all possible propositions (which is a very good notion of a “degenerative” universal grammar, i.e., based on archetypal decomposition, rather than recursive generation) in superposition of factorial involutes mapped as lattices on the decomposed single-valued multiple Riemann surface sheets. We believe these lattices are “Regge lattices”, and the involved propositional calculi, Wheeler’s “pregeometry”.
Gödel did not prove that arithmetic is more fundamental than logic; he proved that arithmetic is more fundamental than 2-valued logic. Gödel did not prove impossible Leibniz’s dream of a universal calculable language; he proved such language (i.e., Musculpt, laser Esperanto, holographic Volopuk, logovisual technology) impossible under 2-valued logic. Just as written mathematical notation is inadequate to transcribe m-logically-valued modes of thought (which require Musculpt as mathematical notation), so proof is no longer an interesting mathematical problem/exercise: no obvious self-evident axioms; no final or first cause; no ultimate decidability, making all decidabilities relative. Proof is a relatively meaningful, but fundamentally meaningless, move in a glass-bead game. Other mathematical activities are far more interesting: injunctions, interrogatories, proclamations, constructions: in the real world, a priori and a posteriori are identity-transparent Kantian categories of unus mundus. The physical universe we are conscious of is the unconscious we strive to bring into consciousness. Not in(t)here is not out(t)here! Non-orientable self-reference is a matter of difficulty only under 2-valued logic. Not only is the thing-in-itself unknowable; not only is there no thing-in-itself to know; but the “itself” purported to be distinguishable from the “thing” to be known is no-thing, knowable or unknowable.
We believe that the rules of dodecaphonic music composition Schoenberg evolved in the second decade of the 20th century (in Vienna with Kandinsky, where the two briefly exchanged the roles of musician and painter) for post-atonal composition can be applied to Gödel encoding of logical propositions, such that the universal grammar, contained in the transfinite set of Gödel numbers on the m-logically-valued reference space, can be displayed/played as Musculpt. We believe Kandinsky's ideas, like “a triangle can only be yellow”, explicated in On the Spiritual in Art (1910), was in direct reaction to Planck’s notion of a quantum of action and an attempt to crystallize the synaesthetic colored-hearing aspects of Musculpt he was soon to paint as universal grammar of form in process, which can be rigorously developed by studying frequency correlates of Gödel numbers which involute into logical propositions arrayed on the multiple sheets as lattices. The harmonic statements plotted on dodecaphonic composition matrices can be stacked, and clearly can be Gödel encoded, which means that an m-logically-valued argument can be played back musically and holo-sculpturally. Or, vice versa, music-sculpture topologies can be decrypted into their equivalent logical propositions. Gödel’s definitions on primes, relative to “equals”, “plus”, “minus”, and so on, thus, need to be empirically verified relative to nature’s form in process. His specific choices for prime number codons were made on basis of the glass-bead game play which concerned him. Other choosing algorithms related to classes of natural process (such as severe storm genesis, DNA generated coherent waves) would establish other sets of prime number codons generating other classes of Gödel numbers (including those generated under m-valued logics and “axiooooomitized” relative to non-self-identical numbers in MOON). But this “direction” of formulation is ego-complex inverted from the unus mundus case. The universal consciousness (“collective unconscious” from perspective of the ego-complex) moves as Musculpt. In order for Musculpt movement to enter (more accurately “congeal”) a brain, trigger neuronal firing patterns, and thus engage in spontaneous localization, it must involutionally Gödel encode: this is why the ancient world was so transfixed by gematria, by stellar configurations (point sets) as astrological propositions, injunctions, interrogatories, proclamations, constructions. Paradoxically, and tragically, the Inca appear to have succumbed to 200 Spaniards by being thus transfixed. The question is, Is there a universal grammar of Gödel encoding for the hierarchy of Russellian types of Gödel numbers?