XPost: comp.theory   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 17/11/2025 00:52, olcott wrote:   
   > On 11/16/2025 1:11 PM, Tristan Wibberley wrote:   
   >> On 16/11/2025 01:29, olcott wrote:   
   >>> G ↔ ¬Prov(⌜G⌝)   
   >>> Directed Graph of evaluation sequence   
   >>> 00 ↔               01 02   
   >>> 01 G   
   >>> 02 ¬               03   
   >>> 03 Prov            04   
   >>> 04 Gödel_Number_of 01  // cycle   
   >>   
   >>   
   >> I would argue that ⊢ G := ¬Prov(⌜G⌝) is not a theorem. Before you   
   find   
   >> that a value of G exists (so that you may evaluate G) you must first   
   >> find that the statement defining G is a theorem.   
   >>   
   >   
   > https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom   
   > It is the G sentence of the first incompleteness theorem.   
   >   
   > ?- G = not(provable(F, G)).   
   > G = not(provable(F, G)).   
   >   
   > ?- unify_with_occurs_check(G, not(provable(F, G))).   
   > false.   
   >   
   > It has a cycle proving that it cannot be resolved   
   > because it remains stuck in an infinite loop.   
      
      
   A proposition that claims G so defined is not a theorem, therefore G is   
   not so defined. It is not a theorem based on proof by contradiction:   
      
      
   Suppose a proposition   
   derive absurdity from it   
   conclude the proposition is not a theorem   
      
      
   Suppose G := not(provable(F, G))   
   derive absurdity (G contradicts itself)   
   conclude G :≠ not(provable(F,G))   
      
      
   We're not about to say that proof by contradiction is invalid.   
      
      
   --   
   Tristan Wibberley   
      
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