XPost: sci.math, comp.theory   
   From: polcott333@gmail.com   
      
   On 1/23/2026 3:13 AM, Mikko wrote:   
   > On 22/01/2026 18:40, olcott wrote:   
   >> On 1/22/2026 2:21 AM, Mikko wrote:   
   >>> On 21/01/2026 17:22, olcott wrote:   
   >>>> On 1/21/2026 3:03 AM, Mikko wrote:   
   >>>>>   
   >>>>> No, it hasn't. In the way theories are usually discussed nothing is   
   >>>>> "ture in arithmetic". Every sentence of a first order theory that   
   >>>>> can be proven in the theory is true in every model theory. Every   
   >>>>> sentence of a theory that cannot be proven in the theory is false   
   >>>>> in some model of the theory.   
   >>>>>   
   >>>>>> only because back then proof theoretic semantics did   
   >>>>>> not exist.   
   >>>>>   
   >>>>> Every interpretation of the theory is a definition of semantics.   
   >>>>>   
   >>>>   
   >>>> Meta‑math relations about numbers don’t exist in PA   
   >>>> because PA only contains arithmetical relations—addition,   
   >>>> multiplication, ordering, primitive‑recursive predicates   
   >>>> about numbers themselves—while relations that talk about   
   >>>> PA’s own proofs, syntax, or truth conditions live entirely   
   >>>> in the meta‑theory;   
   >>>   
   >>> Methamathematics does not need any other relations between numbers   
   >>> than what PA has. But relations that map other things to numbers   
   >>> can be useful for methamathematical purposes.   
   >>>   
   >>>> so when someone appeals to a Gödel‑style relation like   
   >>>> “n encodes a proof of this very sentence,” they’re   
   >>>> invoking a meta‑mathematical predicate that PA cannot   
   >>>> internalize, which is exactly why your framework draws   
   >>>> a clean boundary between internal proof‑theoretic truth   
   >>>> and external model‑theoretic truth.   
   >>>   
   >>> Anyway, what can be provven that way is true aboout PA. You can deny   
   >>> the proof but you cannot perform what is meta-provably impossible.   
   >>   
   >> Gödel’s sentence is not “true in arithmetic.”   
   >> It is true only in the meta‑theory, under an   
   >> external interpretation of PA (typically the   
   >> standard model ℕ). Inside PA itself, the sentence   
   >> is not a truth‑bearer at all.   
   >   
   > There is no concept of "truth-bearer" in an uninterpreted theory because   
   > there is not concept of "truth". The relevant concept is "sell-formed-   
   > formula" and Gödels sentence is one. It may be true or false in an   
   > interpretation.   
   >   
      
   There is a   
   "true on the basis of meaning expressed in language"   
   and I figured out how to make it computable over the   
   body of knowledge.   
      
   > Gädel's metatheory contains PA. In Gödel's interpretation PA is   
   > interpreted in the same way as the PA part of the metathoéory.   
   > Gödel proves that G of PA as interpreted in the metatheory is   
   > true but cannot be proven in PA.   
   >   
      
      
   --   
   Copyright 2026 Olcott

   
      
   My 28 year goal has been to make 
   
   "true on the basis of meaning expressed in language"
   
   reliably computable.

   
      
   This required establishing a new foundation
   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)