[continued from previous message]   
      
   >>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:   
   >   
   >- 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   
      
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