Double-exponentials may be single-valued decidable only when sufficiently smoothed with a little incremental Newtonian nudging, which may mean we’ve entered a region awash in Gibbs oscillations. Will we ever find a way to escape false numerical dispersion? The magnitude of the diffusion operator is being affected by high frequency ripples. We’re going to be systematically truncated! Can a positive-definite advection scheme - in flux form - coupled to split-explicit time integration possibly save us from the inevitable discontinuities associated with strong initialization shock? Differential equation sets and wave functions with unsmooth initial conditions and data may present intractable problems, precluding unique solution, i.e., single-valued outcomes are not prescribed. Ill-posed problems may be Turing-uncomputable: the proposition that prediction requires detailing of the spatial field recursively (rather than description of an elemental oscillator), such that an infinite sequence of computational operations is identified with an infinite series of absolutely-in-so-far-as-distinct selfsame instants of linear-time (in violation of principles devolving from the continuum hypothesis), is ill-posed, thus precluding computability, particularly with binary processors.
I-Derek began reading into Gödel’s theorem in the early 60s, about the time he met Ernest Nagel at the Sheraton Park Hotel where the philosopher of science was to have moderated a debate between George and McGeorge - Kahin and Bundy, that is. Bundy didn’t show, however, at this First National Teach-in Against the Vietnam War. Nagel lived in Northwest Washington, D.C., and was married to a very accomplished Jungian analyst. They had a massive library in a modest home near the zoo - and discussions there were elaborate, to say the least. Nagel had recently co-authored a book on Gödel. Dinner over Jung and Gödel had been awe inspiring and inspirational to a mere undergraduate (soon to become a college drop-out). And I-Derek never stopped reading about Gödel. But the more he learned, the more ludicrous it all seemed, indeed, pathetic: all that clinging to security blankets like single-valued consistency, recursive calculability, simply-connected and orientable completeness, for godsake, and so on. What was important about Gödel’s theorem is not the single-valued-logic consistency/completeness issue upon which so much attention has been lavished, but the very creation of Gödel numbers (which stand for metamathematical statements, statements which could be composed according to rules of dodecaphonic music composition, thus laying a foundation for Musculpt: suggested in THE MOON OF HOA BINH through a self-reentrant poem based on a retrograde inversion of a tonal sequence given in the Vietnamese language, in which a cryptogram related to m-valued identity is hidden as the primary clue needed to unravel a murder mystery) by which Gödel obtained his results. Machine languages of m-valued quantum processors will be based upon Gödel-number arithmetics and algebras. All integers are Gödel numbers because there is no statement that does not make sense in some order of logical-value (relative to Post’s m-valued logics). Prevailing forms of use in any given cultural context would limit utility of some classes of statement, but this limitation is not equivalent to meaninglessness.
Turing was brilliantly moving in the same domains as Gödel when impressed into war service, where the weight of strategic urgency forced choices he otherwise would never have made: reject infinite sets; embrace infinite sequences; calculate only via finite sums. Thus, was the worldline of computer science determined by the needs of war. Computers were created in a hurry to solve a life-and-death problem; they were not created out of the full depth of thought Turing was capable of and had been vectored upon before being suborned to war. This was the more probable bedrock of suicide: no route of return to his personal cognitive status quo ante. I-Derek determined he would never be another Turing - even as he went off to the Vietnam War. Had Turing been given his own head, the first computers would have been quantum computers, not recursive machines - quantum computers processing via interactive wave functions, each with superposed transfinite sets of actual m-valued relative states. Tell me, is completeness really an interesting issue under the continuum hypothesis?