From: mikko.levanto@iki.fi   
      
   On 01/02/2026 17:18, olcott wrote:   
   > On 2/1/2026 4:28 AM, Mikko wrote:   
   >> On 31/01/2026 17:23, olcott wrote:   
   >>> On 1/31/2026 2:41 AM, Mikko wrote:   
   >>>> On 30/01/2026 16:35, olcott wrote:   
   >>>>> On 1/30/2026 3:34 AM, Mikko wrote:   
   >>>>>> 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.   
   >>>>>   
   >>>>> It would only seem to have no effect because you   
   >>>>> never bothered to understand what an occurs_check is.   
   >>>>   
   >>>> That assumption is false.   
   >>   
   >>> So far you have conclusively proven that you   
   >>> do not understand what an occurs_check is.   
   >>   
   >> That's false. Your "proof" is not sound.   
   >>> If you want to provide that you do know then   
   >>> you must provide all of the correct details.   
   >>   
   >> That's false. Irrelevant details should not be included. Obvious details   
   >> shold not be included, either, except those that someone asks about.   
   >>   
   >>> Merely claiming that my statement is false   
   >>> is an assertion entirely bereft of supporting   
   >>> reasoning thus inherently baseless.   
   >>   
   >> If you don't understand some point in the justification you may ask.   
   >   
   > baseless claims are rejected out-of-hand   
      
   You falsely call baseless justifications that you want to reject   
   but can't find refutation or can't even understand.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)