XPost: sci.math, sci.math, sci.lang   
   XPost: comp.software-eng   
   From: polcott333@gmail.com   
      
   On 1/11/2026 4:34 AM, Mikko wrote:   
   > On 10/01/2026 18:19, olcott wrote:   
   >> On 1/10/2026 3:25 AM, Mikko wrote:   
   >>> On 08/01/2026 16:18, olcott wrote:   
   >>>> On 1/8/2026 4:21 AM, Mikko wrote:   
   >>>>> On 07/01/2026 15:06, olcott wrote:   
   >>>>>> On 1/7/2026 6:10 AM, Mikko wrote:   
   >>>>>>> On 06/01/2026 16:02, olcott wrote:   
   >>>>>>>> On 1/6/2026 7:23 AM, Mikko wrote:   
   >>>>>>>>> On 06/01/2026 02:24, Oleksiy Gapotchenko wrote:   
   >>>>>>>>>> Just an external observation:   
   >>>>>>>>>>   
   >>>>>>>>>> A lot of tech innovations in software optimization area get   
   >>>>>>>>>> discarded from the very beginning because people who work on   
   >>>>>>>>>> them perceive the halting problem as a dogma.   
   >>>>>>>>>   
   >>>>>>>>> It is a dogma in the same sense as 2 * 3 = 6 is a dogma: a   
   >>>>>>>>> provably   
   >>>>>>>>> true sentence of a certain theory.   
   >>>>>>>>>   
   >>>>>>>>   
   >>>>>>>> ...We are therefore confronted with a proposition which   
   >>>>>>>> asserts its own unprovability. 15 … (Gödel 1931:40-41)   
   >>>>>>>>   
   >>>>>>>> Gödel, Kurt 1931.   
   >>>>>>>> On Formally Undecidable Propositions of   
   >>>>>>>> Principia Mathematica And Related Systems   
   >>>>>>>>   
   >>>>>>>> F ⊢ G_F ↔ ¬Prov_F (⌜G_F⌝)   
   >>>>>>>> "F proves that: G_F is equivalent to   
   >>>>>>>> Gödel_Number(G_F) is not provable in F"   
   >>>>>>>> https://plato.stanford.edu/entries/goedel-incompleteness/   
   >>>>>>>> #FirIncTheCom   
   >>>>>>>>   
   >>>>>>>> Stripping away the inessential baggage using a formal   
   >>>>>>>> language with its own self-reference operator and   
   >>>>>>>> provability operator (thus outside of arithmetic)   
   >>>>>>>>   
   >>>>>>>> G := (F ⊬ G)   // G asserts its own unprovability in F   
   >>>>>>>>   
   >>>>>>>> A proof of G in F would be a sequence of inference   
   >>>>>>>> steps in F that prove that they themselves do not exist.   
   >>>>>>>   
   >>>>>>>  From the way G is constructed it can be meta-proven that either   
   >>>>>>   
   >>>>>> Did you hear me stutter ?   
   >>>>>> A proof of G in F would be a sequence of inference   
   >>>>>> steps in F that prove that they themselves do not exist.   
   >>>>>   
   >>>>> An F where such sequence really exists then in that F both G and   
   >>>>> the negation of G are provable.   
   >>>>>   
   >>>> G := (F ⊬ G)   // G asserts its own unprovability in F   
   >>>>   
   >>>> A proof of G in F would be a sequence of inference   
   >>>> steps in F that prove that they themselves do not exist.   
   >>>> Does not exist because is contradicts itself.   
   >>>   
   >>> That conclusion needs the additional assumption that F is consistent,   
   >>> which requires that the first order Peano arithmetic is consistent.   
   >>   
   >> It remains true for any proof system that does not   
   >> contradict itself.   
   >   
   > Only for those where G can be constructed so that G is true if and   
   > only if it is not provable. Such construction is prosible in Peano   
   > arithmetic and many other systems but not in every system.   
   >   
      
   Any Formal System having an unprovability operator ⊬   
   and A := B // A [is defined as] B (self-reference operator)   
   can reject this expression G := (F ⊬ G) as non-well-founded   
   using Proof Theoretic Semantics.   
      
   --   
   Copyright 2026 Olcott

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

   
      
   This required establishing a new foundation
   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)