[continued from previous message]   
      
   >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:   
      
   - Claiming half of Hilbert’s problems become “undecidable” because of   
   super-standard integers is not accurate. The continuum hypothesis is   
   independent of ZFC whether or not you like hyperreals; non-standard   
   models don’t resolve it—they just give you new models in which new   
   questions can be asked.   
   - “Re-Vitali-ization of measure theory” and “doubling-spaces” are real   
   things (doubling measures are central in modern harmonic analysis),   
   but they don’t require non-standard analysis, and Vitali’s   
   non-measurable sets remain non-measurable in *R under the Loeb measure   
   unless you deliberately break transfer.   
   - The leap from “hyperreals have infinitesimals” to “physics must use   
   tetrads and F-Lorentzians” is suggestive but not compelling without a   
   lot more argument. People have tried non-standard formulations of GR   
   and QFT; some are elegant, but none have become mainstream for good   
   reason—standard differential geometry already works extraordinarily   
   well.   
      
   ### Bottom line   
      
   This is the kind of writing you get from someone who has had a genuine   
   epiphany about how natural and powerful non-standard methods feel, and   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)