XPost: sci.math, comp.theory   
   From: mikko.levanto@iki.fi   
      
   On 27/01/2026 17:29, olcott wrote:   
   > On 1/27/2026 2:15 AM, Mikko wrote:   
   >> On 26/01/2026 18:45, 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?   
   >   
   > A meta level is even required to prove what is true   
   > in actual PA. This meta-level does a back-chained   
   > inference from x to the axioms of PA.   
      
   The expession "true in PA" is usually not used. In uninterpreted PA   
   the sentences that can be proven without additional assumptions are   
   theorems and the rest are not.   
      
   >> In the metatheory one can construct a model of PA. Everyting in that   
   >> model can be proven in the metatheory is true in that model so it is   
   >> true in some model of PA.   
   >   
   > Yes I am making sure to exclude that.   
      
   Exclude what? That a proven sentence can be true?   
      
   --   
   Mikko   
      
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