XPost: comp.theory, sci.math   
   From: mikko.levanto@iki.fi   
      
   On 27/01/2026 16:48, olcott wrote:   
   > On 1/27/2026 2:05 AM, Mikko wrote:   
   >> 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.   
   >   
   > Mendelson simply uses ⊢ 𝒞 to indicate that 𝒞 is a theorem.   
      
   That is the usual metalogical notation.   
      
   > ∀x (True(x) ≡ ⊢ 𝒞)   
      
   Usually the symbol True, if used at all, is reserved for other purposes.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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