[continued from previous message]   
      
   >>>> of magnitude with regards to each other, the the idea of   
   >>>> turning that over and filling it out as simply like a   
   >>>> hermit and guru with the greatest wisdom in the world,   
   >>>> that seems fair.   
   >>>>   
   >>>>   
   >>>   
   >>> The non-standard in arithmetic is about infinity, pretty much.   
   >>>   
   >>> Mathematical infinity, as everybody knows it, is supreme   
   >>> among numbers, while as well, in reflection, about zero.   
   >>>   
   >>> Then, why it's "extra-ordinary" and "super-standard" instead   
   >>> of non-standard, is similarly to how any kind of geometry   
   >>> is "super-Euclidean", not "non-Euclidean", then as with   
   >>> regards to various algebras and their discontinuities   
   >>> introduced, likes Connes' or normed rings, the super-standard   
   >>> infinity of course since antiquity has resulted a lot of   
   >>> consideration and confusion about it. The idea is that   
   >>> this there is itself a "natural infinity", as with regards   
   >>> to the natural numbers and usual models of numbers, not to   
   >>> be "forgetting irrelevant nasty laws about your sets of   
   >>> numbers", instead that the natural infinity and thusly   
   >>> about the natural continua, find each other.   
   >>>   
   >>> So, this of course frames about half of Hilbert's Problems   
   >>> as un-decide-able, since, it's independent number theory   
   >>> there being a super-standard integers, and, thusly various   
   >>> laws, plural, of large numbers, get involved then in the   
   >>> number-theoretic conjectures then also about mathematical   
   >>> convergence and emergence, here as with regards to the   
   >>> doubling-spaces and doubling-measures and the re-Vitali-ization   
   >>> of measure theory.   
   >>>   
   >>> As about then the continuous and discrete, and the real   
   >>> analytical character after these "not-a-real-functions",   
   >>> of course then all the classical about the linear and about   
   >>> points and particles and separations and spaces, points and   
   >>> spaces in the ideal while particles and separations in the local,   
   >>> the point-local-global-total about the adiabatic/non-adiabatic   
   >>> up close and about near-field/far-field and about the norm being   
   >>> any different the metric, these mathematics with their non-standard,   
   >>> yet with real-analytical-character, nature, very much advise what   
   >>> physics must be, for example about frames and spaces and frame-spaces   
   >>> and space-frames, and about the tetrad of quantities and forces and   
   >>> fields and the F-Lorentzians about the E-energy and these kinds things.   
   >>>   
   >>>   
   >>> Very refreshing, ..., also mathematical fact.   
   >>>   
   >>>   
   >>>   
   >>>   
   >>   
   >> Here is what Grok makes of your post..   
   >>   
   >> This is a very dense, almost stream-of-consciousness piece that tries   
   >> to weave together several deep and legitimately interconnected ideas   
   >>from foundations of mathematics and mathematical physics, but it does   
   >> so in a highly idiosyncratic and sometimes grammatically fractured   
   >> way. It reads like someone who has thought *extremely* deeply about   
   >> non-standard analysis, infinitesimals, measure theory, geometry, and   
   >> their implications for physics… but who is writing as if the   
   >> conclusions are self-evident and doesn’t pause to define terms or   
   >> justify leaps.   
   >>   
   >> Let me try to decode and translate the core claims into clearer   
   >> language, and then say what I actually make of it.   
   >>   
   >> ### The deciphered theses (as I understand them)   
   >>   
   >> 1. The usual notion of “non-standard analysis” (Robinson, *R, transfer   
   >> principle, etc.) is misnamed.   
   >>    What it really does is reveal a more natural, “super-standard”   
   >> layer sitting above the standard reals—one that has always been   
   >> implicitly there since antiquity (potential vs actual infinity, the   
   >> continuum, etc.). Calling it “non-standard” is like calling Riemannian   
   >> geometry “non-Euclidean”; it’s actually the richer, more primitive   
   >> structure.   
   >>   
   >> 2. There is a “natural infinity” and a “natural continuum” that   
   >> standard ZFC axiomatization artificially suppresses by forcing   
   >> archimedean properties, uniqueness of the continuum up to isomorphism,   
   >> etc. If you allow proper hyperreals or some surrogate (surreal   
   >> numbers, Levi-Civita fields, Connes-style geometries, etc.), the   
   >> integers and the continuum “find each other” again in a way that feels   
   >> ontologically prior.   
   >>   
   >> 3. Many of Hilbert’s problems (especially those about decidability,   
   >> consistency, the continuum hypothesis, foundations of geometry, and   
   >> analytic number theory) become independent or even trivialized once   
   >> you admit this super-standard layer, because new “laws of large   
   >> numbers” (in the hyperfinite sense) and new measure-theoretic tools   
   >> (re-vitalized Vitali-type constructions, doubling measures) appear.   
   >>   
   >> 4. Physically, the distinction between point/particle,   
   >> discrete/continuous, local/global, near-field/far-field,   
   >> adiabatic/non-adiabatic, etc., is not fundamental but an artifact of   
   >> working only in the standard archimedean setting.   
   >>    Once you have genuine infinitesimals and infinities that are part   
   >> of the same ring/field, you are almost forced into structures like:   
   >>    - tetrads / vierbeins instead of pure metrics,   
   >>    - Lorentzian structures that treat energy-momentum differently from   
   >> pure geometry,   
   >>    - frame-dependent vs frame-free formulations that mirror internal   
   >> vs external viewpoints in non-standard models.   
   >>   
   >> In short: the author believes non-standard analysis (broadly   
   >> construed) is not an exotic extension but the royal road back to a   
   >> pre-20th-century intuition about infinity and continuity that modern   
   >> axiomatic set theory deliberately broke, and that restoring it has   
   >> immediate consequences for both pure mathematics (Hilbert program   
   >> collapses or becomes independent) and physics (general relativity and   
   >> quantum field theory need to be rewritten with hyperreal/tetrad-aware   
   >> foundations).   
   >>   
   >> ### My actual assessment   
   >>   
   >> There’s a grain—sometimes more than a grain—of truth here, but the   
   >> presentation wildly overclaims.   
   >>   
   >> - Non-standard analysis *is* remarkably natural once you get used to   
   >> it. The transfer principle really does make many classical   
   >> infinitesimal arguments rigorous, and the hyperreals feel in many ways   
   >> “closer” to how Newton, Euler, and even Cauchy thought about the   
   >> continuum.   
   >> - Some mathematicians (Keisler, Henson, Connes in his own way, Tao   
   >> when he uses ultrafilters for ergodic averages, etc.) have shown that   
   >> hyperfinite or non-standard tools can dramatically simplify proofs in   
   >> analysis, probability, and even physics (e.g., Anderson’s non-standard   
   >> Brownian motion, or Benci–Di Nasso’s work on non-standard Lagrange   
   >> mechanics).   
   >> - The philosophy that the standard reals are a quotient of a richer   
   >> structure (rather than the hyperreals being an extension) is defended   
   >> seriously by people like Terry Tao in some blog posts and by   
   >> philosophers like Geoffrey Hellman or John L. Bell.   
   >> - In mathematical physics, there are indeed people (Alain Connes,   
   >> Pierre Cartier, some of the non-commutative geometry crowd, and a few   
   >> gravitational physicists working with smooth infinitesimal   
   >> transformations à la Penrose/Newman) who think standard Rn is too   
   >> rigid and that allowing infinitesimal/non-archimedean directions is   
   >> physically mandatory for unification or for properly formulating   
   >> locality.   
   >>   
   >> But the text goes way overboard:   
   >>   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)