XPost: comp.theory, sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   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.   
      
   Did he really mean that there's some level of completeness in which   
   there is meaninglessness (things that look like propositions but which   
   are not? Well, duh. But arithmetic isn't required for that, merely   
   self-references such as non-ranked definitions and fixed-point   
   combinators (the meaning depends on a meaning that depends on a meaning   
   that...).   
      
   Hang on, he had two incompleteness theorems and a completeness theorem.   
   Can we get some good terminology that distinguishes them because I think   
   there's some referential ambiguity creeping in.   
      
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   Tristan Wibberley   
      
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