XPost: comp.theory, sci.math   
   From: mikko.levanto@iki.fi   
      
   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.   
      
   >>> Truth is computable because “meaningful sentence”   
   >>> is defined as “sentence with a well-founded   
   >>> justification tree,” and evaluating any well-founded   
   >>> tree always terminates. Anything else isn’t truth-apt.   
   >>   
   >> That "bcause" is wrong. Whether a sentence has a well-founded   
   >> justifiation tree is not computable, especially for arithmetic   
   >> sentences.   
   >   
   > My one half page of text explaining all of the key details   
   > of my 28 years of work was completely validated by five   
   > different LLM systems. proof theoretic semantics is correct   
   > model theoretic semantics is profoundly wrong-headed.   
      
   That "proof theoretic semantics is correct model theoretic semantics"   
   may indeed be profoundly wrong-headed but there is another possibility   
   that you just don't understand it.   
      
   > Your ignorance of the details of well-founded proof theoretic   
   > semantics makes your rebuttal baseless.   
      
   No, that does not follow.   
      
   >> But that does not alter the fact that an existence or non-existence   
   >> of a metalanguage feature that is not present in the justification   
   >> tree is irrelevant.   
      
   > An existence or non-existence of a metalanguage feature   
   > is entirely anchored in a totally wrong-headed notion.   
      
   It does not matter where sometihing irrelevant is anchored.   
      
   > The only way that this can be seen is to become an expert   
   > in well-founded proof theoretic semantics.   
      
   No reason to belive that.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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