XPost: comp.theory, sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 03/01/2026 21:39, Mike Terry wrote:   
   > On 03/01/2026 19:02, Tristan Wibberley wrote:   
   >> On 03/01/2026 16:32, Mike Terry wrote:   
   >>> 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.   
   >>>   
   >>> Neither G nor ¬G has a derivation (in your terms, a "formal prooof")   
   >>> within the base system.  That is what Godel proves, showing that the   
   >>> base system is incomplete.   
   >>   
   >> That can't be what he meant can it? Lots of systems were known to have   
   >> statements that had no derivation, all nonsense statements, for example.   
   >   
   > Yes it was what he meant!  :/   
   >   
   > His theorem was about formal systems of arithmetic.  Such systems don't   
   > contain "nonsense statements".  They have construction rules that define   
   > what constitues a well formed formula (WFF), and amongst those what so   
   > constitutes a "sentence".  The semantics for the system define what   
   > every sentence "means".  It is not possible to create "nonsense" sentences.   
      
   Thanks, his 1931 paper isn't very clear about what, exactly, it is   
   demonstrating.   
      
   The G ≡ ¬Provable(G) isn't terribly remarkable although I still haven't   
   got the the bottom of the meaning of that because I still rely on   
   ontology for "Provable" but I'm sure I will think it through. I wonder   
   if it turns out that a real "provability" object which has its natural   
   meaning the same as its properties just can't be encoded in a formal   
   system and that's the real meaning of "incomplete" (ie, putting its   
   properties in the primitive frame gives an inconsistent system).   
      
   How does he demonstrate that system P has the amount of arithmetic   
   claimed to force an incomplete system *and* demonstrate that there is   
   nothing else in the system which could cause it in combination with   
   arithmetic? I suspect it's really about the embedding that forms the   
   meta-system: "forall such-a-class-of embeddings of arithmetic ..."   
      
   --   
   Tristan Wibberley   
      
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