XPost: sci.math, comp.theory, sci.math.symbolic   
   From: news.x.richarddamon@xoxy.net   
      
   On 1/10/26 11:19 AM, 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 nexist.   
   >>> 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.   
   >   
   >> If F is not consistent then both G and its negation are provable in F.   
   >> The first order Peano arithmetic is believed to be sonsistent but its   
   >> consistency is not proven.   
   >>   
   >   
   > The point is that after all these years no one ever   
   > bothered to notice WHY G is unprovable in F. When   
   > we do that then Gödel Incompleteness falls apart.   
   >   
   > *G is unprovable in F because its proof would contradict itself*   
   > *G is unprovable in F because its proof would contradict itself*   
   > *G is unprovable in F because its proof would contradict itself*   
   >   
   >   
      
   Right. so you can only have two of the following, and not all three:   
      
   1) Consistent.   
   2) Complete   
   3) Capable of supporting the Natural Numbers.   
      
   It seems the logic you can handle can't do the last, so yo are fine with   
   your limited, but Complete and Consistant system.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)