XPost: sci.math, comp.theory   
   From: news.x.richarddamon@xoxy.net   
      
   On 1/28/26 1:08 PM, olcott wrote:   
   > 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.   
      
      
   In other wor4s, you admit you don't know what you are talking about.   
      
   I guess you just don't know what "logic" is.   
      
   >   
   >> ...   
   >>   
   >>>> ∀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.   
      
   Except it isn't, as Tarski showed. Once your system is as powerful as   
   PA, which means it can handle "Godel Arithmatic" as a method of creating   
   symantics, the existance of a Truth Predicate just makes the systme   
   inconsistant.   
      
   Your problem is you just don't understand what "semantics" actually mean.   
      
   >   
   > *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?   
   >>   
   >   
   >   
      
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