XPost: comp.theory, sci.math   
   From: mikko.levanto@iki.fi   
      
   On 27/01/2026 17:32, olcott wrote:   
   > On 1/27/2026 2:17 AM, Mikko wrote:   
   >> On 26/01/2026 18:58, olcott wrote:   
   >>> On 1/26/2026 10:45 AM, Richard Damon wrote:   
   >>>> On 1/26/26 10:22 AM, olcott wrote:   
   >>>>> On 1/26/2026 6:55 AM, Mikko wrote:   
   >>>>>> On 25/01/2026 15:30, olcott wrote:   
   >>>>>>> On 1/25/2026 5:24 AM, Mikko wrote:   
   >>>>>>>> On 24/01/2026 16:18, olcott wrote:   
   >>>>>>>>> On 1/24/2026 2:23 AM, Mikko wrote:   
   >>>>>>>>>> On 22/01/2026 18:47, olcott wrote:   
   >>>>>>>>>>> On 1/22/2026 2:21 AM, Mikko wrote:   
   >>>>>>>>>>   
   >>>>>>>>>>>> Anyway, what can be provven that way is true aboout PA. You   
   >>>>>>>>>>>> can deny   
   >>>>>>>>>>>> the proof but you cannot perform what is meta-provably   
   >>>>>>>>>>>> impossible.   
   >>>>>>>>>>   
   >>>>>>>>>>> The meta-proof does not exist in the axioms of PA   
   >>>>>>>>>>> and that is the reason why an external truth in   
   >>>>>>>>>>> an external model cannot be proved internally in PA.   
   >>>>>>>>>>> All of these years it was only a mere conflation   
   >>>>>>>>>>> error.   
   >>>>>>>>>>   
   >>>>>>>>>> It is perfectly clear which is which. But every proof in PA is   
   >>>>>>>>>> also   
   >>>>>>>>>> a proof in Gödel's metatheory.   
   >>>>>>>>>   
   >>>>>>>>> ∀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)))   
   >>>>>>>>   
   >>>>>>>> Those sentences don't mean anything without specificantions of a   
   >>>>>>>> language and a theory that gives them some meaning.   
   >>>>>>>   
   >>>>>>> In other word you do not understand standard notational   
   >>>>>>> conventions that define True for PA as provable from the   
   >>>>>>> axioms of PA and False for PA as refutable from the axioms   
   >>>>>>> of PA.   
   >>>>>>   
   >>>>>> There are no notational convention that defines True, False, and   
   >>>>>> WellFounded with two arguments. And you did not specify in which   
   >>>>>> context your sentences are true or otherwise relevant.   
   >>>>>>   
   >>>>>   
   >>>>> “x is a single finite string representing   
   >>>>> a PA‑formula, such as ‘2 + 3 = 5’.   
   >>>>> True(PA, x), False(PA, x), and WellFounded(PA, x)   
   >>>>> are meta‑level unary predicates classifying   
   >>>>> that formula by its provability in PA.”   
   >>>>>   
   >>>>   
   >>>> In outher words, you ACCEPT that the meta level can define what is   
   >>>> true in PA?   
   >>>>   
   >>>> I thought you said that PA had to be able to determine the truth   
   >>>> itself?   
   >>>   
   >>> We need a meta-level truth predicate anchored   
   >>> only in the axioms of PA itself and thus not   
   >>> anchored in the standard model of arithmetic.   
   >>   
   >> That predicate is not computable.   
   >   
   > That was Tarski's mistake.   
      
   No, Tarski's proof is about a different problem, though the results   
   are related and there are much similarity in the proofs. Tarski did   
   not use Turing machines in the proof but a computability proof must   
   use that.   
      
   > All of the expressions where True(L, x) is not computable   
    > x is semantically incoherent or outside of the domain of knowledge.   
      
   Computability does not depend on semantics or knowledge.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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