XPost: comp.theory   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 29/12/2025 13:37, Richard Damon wrote:   
      
   > Incompleteness is a property of a given Formal System, it says that   
   > there exist a statement that is true in that system, but can not be   
   > proven in that system.   
   >   
   > Godel's proof build a meta-system that constructs a statement that   
   > exists in the base system, and which is true in that base system, but   
   > can not be proven in the base system, only the meta-system.   
      
   What do you mean by "proven" here. Do you mean "derived" ? Normally we   
   say a proposition is true in a system when we mean the proposition is a   
   theorem of the system; it has a derivation in the system: a finite   
   sequence of statements that each is derivable from the axioms and   
   earlier statements in the sequence by application of a deductive rule of   
   the system--where the deductive rules are transitive so that's also   
   application of /some/ deductive rules--and which ends with the   
   proposition that's being derived.   
      
   If there is a derivation then it is provable in the base system, if   
   there /isn't/ a derivation then it is /not/ true in that base system   
   (which is different from saying its contrapositive is true in that base   
   system).   
      
      
   --   
   Tristan Wibberley   
      
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