XPost: comp.theory, sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 02/01/2026 02:45, André G. Isaak wrote:   
   > 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.   
      
   We have to avoid "proven in a particular system" and choose "Derived in   
   a particular system" or "Derived of a particular system" or, since it's   
   well defined, "Theorem of a particular system".   
      
   The problem with "prove" is there are numerous episystems (HA being   
   popular) that provide for "proofs" of statements of systems they're   
   applied to. Technically, episystems may or may not prove the same set of   
   statements that are theorem's of the system they're applied to.   
      
      
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   Tristan Wibberley   
      
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