XPost: comp.theory, sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 03/01/2026 03:30, Richard Damon wrote:   
   > On 1/2/26 9:43 PM, André G. Isaak wrote:   
   >> 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.   
   >   
   > Right, And the statement og Godel's G can be fully derived in the base   
   > system, as it is purely a mathematical relationship using the operations   
   > derivable in the system.   
   >   
   > The implications of this statement can't be understood in the system,   
   > but that isn't a requirment to be a Theorem.   
   >   
   >>   
   >> 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.   
   >   
   > Right, but G isn't like this.   
      
   For Goedel's system of statement quoting (goedel numbering) there's the   
   gotcha where the normally allowable informality of naming a statement   
   with a name that isn't an indeterminate of the system is a problem... it   
   doesn't have a quoted form.   
      
   You can't use that informality and you have to generate a new system   
   with an accommodatingly larger system of quotation (with at least one   
   indeterminate accommodated) or else use the already quotable expression   
   of the statement.   
      
   This way, you find some of the exemplary statements you would have   
   admitted are actually not part of the system because they'd be   
   non-constructive (Olcott occurs here).   
      
   A larger system of quotation might invalidated conclusions,   
   necessitating that they be converted to more contingent ones.   
      
   --   
   Tristan Wibberley   
      
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