XPost: comp.theory, sci.math   
   From: agisaak@gm.invalid   
      
   On 2026-01-01 20:09, Richard Damon wrote:   
   > On 1/1/26 9:45 PM, 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.   
   >   
   > I guess it depends on your definition of a "Theorem".   
   >   
   > I am using the one that goes:   
   >   
   > "A Theorem is a statement that has been proven."   
    >   
   > note, no restriction that the proof was in the system the Theorem is   
   > stated in, as long as the proof shows that it is actually True in that   
   > system.   
      
   A theorem is a statement that can be derived from the axioms of a   
   particular system. It may be true in other systems, but it is only a   
   theorem in systems in which it can be derived.   
      
   An obvious example to illustrate this would be the fact that there are   
   many theorems which can be derived in Euclidean geometry, but which are   
   not theorems of various non-Euclidean geometries. That is to say, not   
   only can they not be derived in those non-Euclidean geometries, but they   
   can be shown to be *false* in those non-Euclidean geometries.   
      
   Theoremhood is always tied to a particular formal system.   
      
   André   
      
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