XPost: sci.logic, sci.math, comp.theory   
   From: polcott333@gmail.com   
      
   On 1/24/2026 1:23 PM, Richard Damon wrote:   
   > On 1/24/26 12:54 PM, olcott wrote:   
   >> On 1/24/2026 11:10 AM, Richard Damon wrote:   
   >>> On 1/24/26 10:44 AM, olcott wrote:   
   >>>>   
   >>>> The statement that G is true and unprovable in PA has   
   >>>> always been counter-factual. It has never actually been   
   >>>> true  PA and that is why it is unprovable in PA.   
   >>>   
   >>> Sure it is. At least it is a FACT that no natural number will   
   >>> statisfy that relationship, and there is no proof in PA of that fact.   
   >>>   
   >>   
   >> Have you ever heard of: "true in the standard model of arithmetic"?   
   >   
   >   
   > Sure, but they are not in Peano Arithmatic, but are (generally) 1st   
   > order variations of the Peano Axioms which lead to alternate number   
   > systems.   
   >   
   > Godel's proof is statd to be in a system with at least the properties of   
   > Peano Arithmatic, having the ability to show the properties of the   
   > "Natural Numbers"   
   >   
   Gödel’s incompleteness theorem only “works” if   
   one smuggles in an external notion of truth   
   (truth in ℕ) and then pretends it is an   
   internal notion of truth (truth in PA).   
   If we refuse to make that identification,   
   incompleteness evaporates.   
      
   --   
   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)