XPost: comp.theory, sci.math   
   From: mikko.levanto@iki.fi   
      
   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.   
      
   --   
   Mikko   
      
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