XPost: sci.math, comp.theory   
   From: polcott333@gmail.com   
      
   On 1/24/2026 2:20 AM, Mikko wrote:   
   > On 23/01/2026 12:22, olcott wrote:   
   >> 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.   
   >   
   > Except that "true on the basis of meaning expressed in language" is   
   > nmt computable and does not cover all of the body of knowldge.   
   >   
      
   When the basis of "true" is proof theoretic semantics   
   internal to the formal system relative to its own axioms   
   and not truth conditional in a separate model outside   
   of the system undecidability ceases to exist.   
      
   --   
   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)