XPost: comp.theory, sci.math, comp.ai.philosophy   
   From: polcott333@gmail.com   
      
   On 1/22/2026 6:23 PM, Tristan Wibberley wrote:   
   > On 20/01/2026 23:08, Tristan Wibberley wrote:   
   >> On 18/01/2026 23:41, olcott wrote:   
   >>   
   >>> I already just said that the proof and refutation of   
   >>> Goldbach are outside the scope of PA axioms.   
   >>   
   >> So Richard is right that you need a truth value for not being covered:   
   >>   
   >> True(S, Goldbach) = OutOfScope   
   >   
   >   
   > Oh ho! but is Goldbach definable as a shortcode for a statement of the   
   > goldbach conjecture in PA? If there's no such statement then it's out of   
   > scope without a truth value for that.   
   >   
      
   Within proof theoretic semantics the lack   
   of a finite proof entails ungrounded thus   
   non-well-founded. My system works over the   
   entire body of knowledge that can be   
   expressed in language. Knowledge excludes   
   unknowns as outside of its domain.   
      
   ∀x ∈ PA ((True(PA, x)  ≡ (PA ⊢ x))   
   ∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~x))   
   ∀x ∈ PA (~WellFounded(PA, x) ≡ (~True(PA, x) ∧ (~False(PA, x))   
      
   > In addition to equality, it requires negation, either forall or exists,   
   > either conjunction or disjunction... but my memory says not all of those   
   > are available in PA in sufficient generality! Oh if only my brain worked   
   > as well as it once did I could work this through in a sitting, instead I   
   > get mentally disorganised.   
   >   
      
      
   --   
   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)