XPost: sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   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?   
      
   ...   
      
   >> ∀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.   
      
   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?   
      
   --   
   Tristan Wibberley   
      
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