XPost: comp.theory, sci.math   
   From: mikko.levanto@iki.fi   
      
   On 29/01/2026 15:57, olcott wrote:   
   > On 1/29/2026 3:12 AM, Mikko wrote:   
   >> On 28/01/2026 15:49, olcott wrote:   
   >>> On 1/28/2026 3:54 AM, Mikko wrote:   
   >>>> 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.   
   >>>   
   >>> Because you refuse to understand the underlying   
   >>> details of what occurs_check means I cannot   
   >>> explain to you how Tarski erred.   
   >>   
   >> Irrelevant. There is no "occurs_check" in Tarski's proof.   
      
   That would have no effet. Even if the metalanguage had an occcurs_check   
   it would not be necessary to use it in a proof.   
      
   --   
   Mikko   
      
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