XPost: sci.math, comp.theory   
   From: polcott333@gmail.com   
      
   On 1/28/2026 10:21 AM, Tristan Wibberley wrote:   
   > On 20/01/2026 05:29, Richard Damon wrote:   
   >> On 1/19/26 9:39 PM, olcott wrote:   
   >>> On 1/17/2026 3:08 PM, olcott wrote:   
   >>>> For nearly a century, discussions of arithmetic have quietly   
   >>>> relied on a fundamental conflation: the idea that   
   >>>> “true in arithmetic” meant “true in the standard model of ℕ.”   
   >>>> But PA itself has no truth predicate, no internal semantics,   
   >>>> and no mechanism for assigning truth values.   
   >   
   > Don't we assume it to be (implicitly) a schematic system, where the   
   > axioms define the deduction rules?   
   >   
      
   That is the conflation error of Gödel's incompleteness.   
   It seems to be saying what you said to casual observers.   
      
   > ...   
   >   
   >>> ∀x ∈ PA ((True(PA, x)  ≡ (PA ⊢ x))   
   >>> ∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~x))   
   >>> ∀x ∈ PA (~TruthBearer(PA, x) ≡ (~True(PA, x) ∧ (~False(PA, x))   
   >>>   
   >>>   
   >>   
   >> PA doesn't have a truth predicate, because it CAN'T.   
   >                    ^^^   
   >          a unary truth predicate   
   >   
   > but perhaps an operation "IsElementaryTheorem_p(system, objects...)"   
   > for each predicate 'p' can be admitted to an extension of PA.   
   >   
      
   You just understand these things more deeply than   
   anyone else here.   
      
   When we refer to Haskell Curry's notion of elementary   
   theorems that are true then anything derived from   
   them is a theorem that is also true. That is the   
   key foundation of proof theoretic semantics:   
      
   "true on the basis of meaning expressed in language"   
   reliably computable for the entire body of knowledge.   
      
   *Please keep comp.theory because I am showing*   
      
   > Perhaps importantly, I note that PA doesn't relate = with ≠ but both   
   > appear in the axioms, naively avoiding the problem of "what do you mean   
   > by 'negation'?" but leaving a problem of "what do you mean by   
   > 'contradiction'?"   
   >   
   > What resolutions do you perceive regarding that?   
   >   
      
      
   --   
   Copyright 2026 Olcott

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

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