some thoughts from the Pensingers
These many years after the “General Process” paper, we are getting better insight into how the multivalued reference space has to be constructed on Hilbert space under mvalued logics. Musculpt and the hierarchy of cs, c's, and c''s are right at the heart of it. If these insights are pulled out of their mlogicallyvalued context and put into 2valued 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 prerequisite to conscious emergence of alwaysthere subliminal Musculpt (which conventions of written notation deny).
Our tornadogenesisrelated speculation that interval spread in the hierarchy of cs, 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 cs, 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 mvalued logics. (Distribution of cs 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 nontrivial solutions (sets of zeros) will disappear.
What will happen is this: on each prime (arrayed on Riemann's line) on the mlogicallyvalued reference sheet will be stacked other primes from the multiplicity of singlelogicallyvalued sheets composing the Riemann surface map of Everett’s universal wavefunction. (In this approach, Cantorian fractal spacetime relates to the stack of singlevalued sheets, which, in turn, relate to Sakharov’s collapse/anticollapse multisheet model of the universe.) On each of the multiplicity of decomposed singlevalued sheets, Riemann’s line will be located differently within the critical band than it is located on the mlogicallyvalued 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 singlevalued logic to logics of mvalues). 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 selfcorrelation (which is anything but a catastrophe! to all those not identified with the egocomplex).
A supradense mlogicallyvalued Hilbertian reference space constructed in this fashion has nonlocality of embedded objects as a fundamental property. Locality is a decomposition issue involving cycles of selfreentry (or, alternatively stated, of cosmological selfforgetting, amnesis  while recomposing the mlogicallyvalued 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 onetoone 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 pointset 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 operatortime. (Operatortime would be characterized by the hypercomplex zeta function, which we believe our canonical equation for harmonic temperature oscillation of
pelectron 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 Gödelencoded segments of Riemann’s critical band (as it meanders the multisheeted Riemann surface) corresponding to invariants of classes of temporal operation on the mlogicallyvalued reference space.
So, were the Gödelnumbered pointset topologies of Gödelencoded mlogicallyvalued propositions to be Riemann surface mapped, every point on the multiple singlevalued sheets would have a number associated with it  some prime, some nonprime. Logical propositions of m1 and less values would be equivalent to lattices of primenumber points, the values of which are multiplied together (such multiplication being very much like neuronal firing pattern sequences mapped on Szentagothai’s multisheeted model of the cerebral cortex). The Gödel numbers resulting from this multiplication would be located on points on the mlogicallyvalued reference space (the most densely pointpacked Riemann surface sheet). The fullyGödelnumbered mlogicallyvalued 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 singlevalued 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 2valued 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 2valued logic. Just as written mathematical notation is inadequate to transcribe mlogicallyvalued modes of thought (which require Musculpt as mathematical notation), so proof is no longer an interesting mathematical problem/exercise: no obvious selfevident axioms; no final or first cause; no ultimate decidability, making all decidabilities relative.
Proof is a relatively meaningful, but fundamentally meaningless, move in a glassbead game. Other mathematical activities are far more interesting: injunctions, interrogatories, proclamations, constructions: in the real world, a priori and a posteriori are identitytransparent 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! Nonorientable selfreference is a matter of difficulty only under 2valued logic. Not only is the thinginitself unknowable; not only is there no thinginitself to know; but the “itself” purported to be distinguishable from the “thing” to be known is nothing, 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 postatonal 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 mlogicallyvalued 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 coloredhearing 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 mlogicallyvalued argument can be played back musically and holosculpturally. Or, vice versa, musicsculpture 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 glassbead 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 mvalued logics and “axiooooomitized”; relative to nonselfidentical numbers in MOON).
But this “direction” of formulation is egocomplex inverted from the unus mundus case. The universal consciousness (“collective unconscious” from perspective of the egocomplex) 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?
What is most interesting to me about Gödel's work is that he chose to ignore Emil Post's mvalued logics, which were on the scene for over a decade when Gödel published his famous proof. This choice on the part of Gödel is roughly on a par with the performance of G. Birkhoff and J. von Neumann at about the same time, i.e., three years prior to the Nazi invasion of Poland, the home ground of mvalued logics. (See: “The Logic of Quantum Mechanics”, Annals of Mathematics, 37, 1936.)
One truly must wonder at the extraordinary lengths the mind is willing to go in order to avoid looking the multivalue straight in the face. Here, Birkhoff and von Neumann recognized that quantum logic has some relation to projective geometry, but where is the Riemann surface stack? Though some doubt is cast on the utility of Hilbert space, where is the recognition that every point in the referencing phasespace is multivalued, that translation across a singlevalued sheet is projected as a static lattice to the multivalued referencing functionspace? How strange that they should invoke the concept of a logical “lattice”, but not view it as a true pointset topology!
These logical lattices of Birkhoff and von Neumann, of course, are of the essence of Post's _{m}T_{m}
logics (1921). Standard AristoteleanBaconianBoolean logic is a _{1}T_{2} logic. Two values are available, one of which is permitted per proposition. This is logic reduced to truthvalue and all the standard syllogistic rules. Post generalized this to the _{m}T_{m} case, where all logical values available are permitted per proposition.
On the Riemann surface stack of singlevalued sheets, there are 1, 2, 3… values available on successive sheets, but only one permitted per proposition (represented by a given logical lattice of connected points). All the values on equivalent points on each such sheet are stacked on their equivalent point on the multivalued reference space (mostdense sheet), where there are mvalues available and mvalues permitted per proposition. Gödel numbering of these mvalued logical lattices necessarily involves expanding the universe of discourse which has grown up around the famed Riemann Hypothesis on distribution of prime numbers (which are factors of Gödel numbers).
With similarly inexplicable myopia, Birkhoff and von Neumann used involutory relations and the concept of skewfields, but the multivalue screaming in the background, without which skewfields are impossible, was completely ignored (along with the whole notion of skewparallelism, qua skewperpendicularity, underlying skewfields, with its profound implications for Riemann's thesis on the origins of charge, Heisenberg's indeterminacy, and resultant nonconservation of energy and virtual phonon exchange underlying mechanisms of high temperature superconductivity, such as that of DNA). Dropping distributive laws for 2valued propositions, indeed! Can laughter be suppressed? B and vN knew of Post's logics, just as did G. These missing recognitions by Birkhoff and von Neumann are not ignorance speaking; they are expressions of psychological dread. And this paper, summarizing at the very least a decade of collective psychoneurotic posturing, was published a mere three years before the inevitable avalanche of consequences!
In arriving at his famous proof, Gödel falsely assumed that: (1) “language A” is orientable, i.e., that “in language A” can be absolutely distinguished from “notin language A”; (2) infinite regress is linear in nature, i.e., that regresses are not selfreentrant marches; (3) description reduces to truth tables, i.e., that there is nothing in logic more fundamental than truthvalue; (4) logical calculi are limited to only one order of logicalvalue, i.e., the 2valued variety; (5) singlevalued propositions are the only legitimate propositions, i.e., that selfcontradiction is a violation of the rules of all logics; (6) completeness is an absolutely meaningful notion, i.e., that there are no languages for which completeness is undefinable  which is not the case for languages involving use of logical calculi with orders of logical value greater than two and propositions with logical values greater than one.
We wonder, What is the Gödel number of this 6fold proposition?
Gödel numbers need to be plotted, not calculated! More accurately stated: factors of mvalued Gödelized propositions need to be plotted on the Riemann surface sheet stack composing the Musculpt manifold, not calculated to supposed single definite values which nonselfidentical numbers do not authentically possess. The collapsedvalue of a superposed number (such as a Gödel number), localized on the Cartesian grid, is not the full measure of this number. Singlevalued Gödelized propositions are a special case (likely a trivial case, cosmologically speaking) of all the mvalued Gödelized propositions there are in this world.
Nonselfidentical superposed numbers are not numerical values! They are FORMS of Musculpt, i.e., archetypes. Arithmetic operations involving such numbers are carried out by operatortime. Nonselfidentical superposed numbers are “dreamtime” configurations of “songlines” (like Riemann's “critical line”). How are such arithmetic operations represented?
Factors of mvalued Gödelized propositions are plotable, even if their multiplied “values” are Turing uncomputable. The resultant plotted lattices are Regge lattices, which are transformable into skewfield curvature configurations by the Regge calculus  which transforms Einstein's field equations into ndimensional lattices, and vice versa.
Where do these lattices underlying the semantic soundedforms of Musculpt fall? Not on the real line. Not on the complex plane. The s (a complex number) of the Euler and Riemann zeta functions is logically 2valued. Its 1/2 + it does not conform to the requirements of _{m}T_{m} as t ranges. The zeta function's s, like the nk landscape of complexity theory, is formulated in terms of ordinary binary logic. The nk landscape is actually a 2^{nk} landscape (for a fuller account of this see: “MValuation in a Generalized Currency Basket”) where the elements in question must be either on or off.
This is a highly relevant comparison because k indicates the number of correlations between the involved elements or factors, n, and, also, because nonselfidentical numbers under mvalued logics are numbers in superposition (i.e., correlated) stacked on a point in the reference space. Generalizing the landscape under _{m}T_{m} yields an M^{nk} landscape.
Were the elements or factors arrayed on this landscape prime factors of Gödel numbers of mvalued propositions, the s of the zeta function would become s^{M}. Now, the power set (set of all subsets of the original set) in Cantor's continuum hypothesis has 2^{n} members, just like the nk landscape of complexity theory has 2^{n} elements or factors. This is a statement about cardinality: the power set has cardinality n.
The power set of nonselfidentical numbers of cardinality n, however, is 2^{nk}  correlations being involved in the nonselfidentical. But this is an incomplete statement, because we are not talking about only the relationships between members of the set (i.e., the many ways in which they can be arrayed into subsets), but about their changing animistic identity transparency under increasing orders of logicalvalue, m. The power set under mvalued logics is not identical to the power set under 2valued logic.
The mlogicallyvalued power set of nonselfidentical numbers of cardinality n, is actually M^{nk}, where M is the order of logicalvalue and k the number of correlation factors (of the involved logical lattice with n bifurcation points). Under such considerations, however, Riemann's zeta s could not be merely a complex number; it would have to be a hypercomplex number, because the complex plane is a _{1}T_{2} logical construct.
Constructs of order _{m}T_{m} are beyond the complex plane. s^{M} is the order of hypercomplexity of s. The mlogicallyvalued power set of nonselfidentical numbers, s^{M}, of cardinality n, is . On this powerset continuum, not Cantor's, can we begin to plot the Gödel number factors of the mvalued logical lattices underlying the soundedforms of Musculpt semantics.
It is only nonselfidentical numbers that make the notion of “cardinality of the continuum” plausible. Degrees of identity transparency, of order m, between the discrete and the continuous, dictated by _{m}T_{m} orders of logic, is the fundamental modification of Peano's axioms required. It can be seen that under Post's logics there is no limit to the number of infinities between that of the natural numbers and that of the continuum, such that, for instance, the “critical line” of Riemann's hypothesis has a unique shadow line under each order of logicalvalue greater than _{1}T_{2}.
Actually, we do not regard consistency an inherent attribute of any “possible world semantic”, except those world semantics of the _{1}T_{2} variety. But, then, I do not regard any possible world semantic as merely possible, for the “possible” _{1}T_{2}variety world semantics are actually treatments of the variety of _{m}T_{m} world semantics as if they were merely possible  merely possible, that is, because consistency is mistakenly considered an inalienable right of world semantics qua world semantics. I must, however, protest that this is no small, purely theoretical, matter in an esoteric corner of one of the myriad fields of logic. If, as regards the 1/2 + it of the Riemann zeta function, the t º a Hermitian operator on Hilbert space, as some have suggested (including Hilbert himself), the “semantic” (import of the involved quantum logic, that is) of some “possible” (in consensus quantum theory, “probable”, however improbable) world is subject to interpretation under _{m}T_{m} such that it becomes iT^{M}, where T would be an mlogicallyvalued Hermitian operator on an mlogicallysheeted Hilbert space. In such a Hilbert space under such a Hermitian operator, the idea of consistency one can entertain becomes a very peculiar notion, indeed. The orders of truth table multiply without bound. Moreover, the doubleexponentials associated with such truthtables indicate they literally turn an imaginary corner (truth table deposed in hyperdimensions) in propositional space, meaning that the i of iT^{M} indicates, not only that the involved numbers have an imaginary component as Riemann's zeta function indicates, but, when given any interpretation whatsoever, so must their resultant contextual meanings have imaginary components: when _{1}T_{2} goes to _{2}T_{m} to _{3}T_{m} to _{4}T_{m} and so on, the imaginary semantic dimensions of the involved worlds multiply hand over foot, as consideration of G. Spencer Brown's calculus of indications would indicate.
In face of unbounded and exponentiated orders of truth table, we personally prefer superposed “actual”, as opposed to separate “possible”, world semantics. Just as EverettWheelerGraham (EWG) argued on basis of lineartimebound thermodynamic considerations that their multiple universes (mapped in a _{1}T_{2}logicconstructed Hilbert space) are incommunicado, in solitary confinement, so Saul Kripke, in constructing world semantics on basis of possibilities, rather than actualities, treats, by direct implication, any meaningintersections between worldsofmeaning as tantamount to trivial: worlds of horsesense, pigsense, dirtyratsense, and so on are overwhelmingly _{1}T_{2} worlds unto themselves. Both the EWG and Kripke cases so transparently justify the individualistic mode of egosphereidentity, caught in inherent alienation and anomie of Existential “separatism”, one is inclined, on that basis alone, to reject them.
Saul Kripke has long been committed to this notion of identity as the only possible world semantic of identity (as a Kantian category). In 1976, if memory serves, Kripke delivered a lecture at Cornell University mistitled in campus promotion as “Time and Eternity”. He began the public lecture by informing his audience that the correct title was “Time and Identity”. Everyone had a good laugh, as all knew no reputable philosopher would be caught these days lecturing on eternity. Some of the more informed in the audience, however, would have been the more happy had the logician lectured on eternity as opposed to identity in relation to lineartime, or more happily yet on eternity in relation to identity, as the monotonic notion of identity he derived (implicitly as a Kantian category) from lineartimeonly so thoroughly neglected the variety of identity constructs experienced historically by the human species, one had to wonder whether or not the promotional snafu was a possible cosmic joke. This possible joke, if it was indeed actual and cosmic, was all the more poignant in that the imaginary corner inevitably turned in propositional truthtable space through doubleexponentials is an imaginary turn away from truthvalue toward animistic identity transparency, which imaginary turn Kripke's derivation of monotonic identity from lineartimeonly implicitly rejected  which rejection, of course, is a foundational assumption of his "possible world semantics", i.e., configurations of meaning arrived at through nonmonotonic reasoning about monotonic identity.
William L. Pensinger & (Cong Huyen Ton Nu) Nha Trang Pensinger
Please check out our Web pages:
http://www.geocities.com/moonhoabinh/
http://www.geocities.com/chtn_nhatrang/
quotes
curiosities
inexplicable secrets of creation
home
contact
