XPost: comp.theory   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 29/12/2025 19:51, Richard Damon wrote:   
   > On 12/29/25 2:21 PM, Pierre Asselin wrote:   
   >> In sci.logic Tristan Wibberley   
   >>  wrote:   
   >>> 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.   
   >>   
   >>> What do you mean by "proven" here. Do you mean "derived" ?   
   >>   
   >> I think Richard misspoke slightly. The undecidable statement is   
   >> true *in the intended interpretation* of the formal system   
   >> (In Goedel's case, the natural numbers with addition and multiplication).   
   >>   
   >> Truth "in the formal system" isn't really defined. You need an   
   >> interpretation.   
   >>   
   >   
   > No, statements in a formal system are DEFINED to be true   
      
   Best to keep pushing the lingufranca toward "to be theorems".   
      
   ps, I note that the use of "defined to be" here is technically wrong but   
   the use of "true" is so much more problematic.   
      
   > if that> statement, referencing object defined in the system model,   
   and related   
   > by relationships defined in the system  can be established starting with   
   > the initial "facts" (axioms) of the system, and following the allowed   
   > logical operations of the system.   
     ^^^^^^^^^^^^^^^^^^   
     ||||||||||||||||||   
     ``````````````````~~~<~~~<~~~< Deduction Rules   
      
      
     A statement in a formal system is included among its theorems exactly   
   when it is derivable from the axioms of the system--which are those of   
   its theorems that are merely supposed--and the derivation is done by the   
   deduction rules of the system.   
      
   We must prepare ourselves mentally for mathematicians to say "in a   
   system" when they mean "in some axiom extension of a system" and to say   
   "is true" when they mean "has a derivable embedding in at least one   
   episystem embedding the system".   
      
   --   
   Tristan Wibberley   
      
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