XPost: comp.theory, sci.math   
   From: mikko.levanto@iki.fi   
      
   On 26/01/2026 17:22, 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.”   
      
   The above is not a notational convention. The symbols may be defined   
   in some context but they are undefined elsewhere.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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