XPost: sci.math, comp.theory, sci.math.symbolic   
   From: mikko.levanto@iki.fi   
      
   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.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)