XPost: comp.theory, sci.math   
   From: polcott333@gmail.com   
      
   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.   
   ∀x (True(x) ≡ ⊢ 𝒞)   
      
      
      
   --   
   Copyright 2026 Olcott

   
      
   My 28 year goal has been to make 
   
   "true on the basis of meaning expressed in language"
   
   reliably computable for the entire body of knowledge.

   
      
   This required establishing a new foundation
   
      
   --- SoupGate-Win32 v1.05   
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