XPost: comp.theory, sci.math   
   From: agisaak@gm.invalid   
      
   On 2026-01-01 16:48, Richard Damon wrote:   
   > On 1/1/26 6:13 PM, Tristan Wibberley wrote:   
   >> On 01/01/2026 22:40, Richard Damon wrote:   
   >>   
   >>> But it IS a theorem of the base system, as it uses ONLY the mathematical   
   >>> operations definable in the base system. What makes you think it isn't a   
   >>> Theorem in the base system.   
   >>   
   >> It has no derivation in the base system, if it had you wouldn't think   
   >> the base system were incomplete.   
   >>   
   >   
   > It has no PROOF in the base system.   
      
   Which means it is not a theorem of the base system. A theorem is a   
   statement which can be proven in a particular system.   
      
   > The statement is surely a statement in the base system.   
      
   Sure, but being a statement and being a theorem are two different things.   
      
   > It is shown to be true there, by a proof in the meta-system.   
      
   It is shown to be true in the base system, but only within the   
   metasystem. Within the base system it cannot be so shown, which   
   precludes it from being a theorem. That's the entire point of Gödel:   
   truth and theoremhood cannot be made to coincide except in very trivial   
   systems.   
      
   > I think you are confused about what you talk about.   
      
   I think you are both somewhat confused here.   
      
   André   
      
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