XPost: comp.theory, comp.software-eng, sci.logic   
   XPost: sci.math   
   From: mikko.levanto@iki.fi   
      
   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   
   G is true and unprovable in F (which means that F is incomplete)   
   or G is false and provable in F (which means that F is inconsistent).   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)