XPost: comp.theory, sci.math   
   From: mikko.levanto@iki.fi   
      
   olcott kirjoitti 4.12.2025 klo 16.06:   
   > On 12/4/2025 2:58 AM, Mikko wrote:   
   >> Tristan Wibberley kirjoitti 4.12.2025 klo 4.32:   
   >>> On 30/11/2025 09:58, Mikko wrote:   
   >>>   
   >>>> Note that the meanings of   
   >>>>   ?- G = not(provable(F, G)).   
   >>>> and   
   >>>>   ?- unify_with_occurs_check(G, not(provable(F, G))).   
   >>>> are different. The former assigns a value to G, the latter does not.   
   >>   
   >>> For sufficiently informal definitions of "value".   
   >>> And for sufficiently wrong ones too!   
   >>   
   >> It is sufficiently clear what "value" of a Prolog variable means.   
      
   > % This sentence cannot be proven in F   
   > ?- G = not(provable(F, G)).   
   > G = not(provable(F, G)).   
   > ?- unify_with_occurs_check(G, not(provable(F, G))).   
   > false.   
   >   
   > I would say that the above Prolog is the 100%   
   > complete formal specification of:   
   >   
   > "This sentence cannot be proven in F"   
      
   The first query can be regarded as a question whether "G = not(provable(   
   F, G))" can be proven for some F and some G. The answer is that it can   
   for every F and for (at least) one G, which is not(provable(G)).   
      
   The second query can be regarded as a question whether "G = not(provable   
   (F, G))" can be proven for some F and some G that do not contain cycles.   
   The answer is that in the proof system of Prolog it cannot be.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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