XPost: comp.theory, sci.logic, sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 18/01/2026 23:41, olcott wrote:   
      
   > I already just said that the proof and refutation of   
   > Goldbach are outside the scope of PA axioms.   
      
   So Richard is right that you need a truth value for not being covered:   
      
   True(S, Goldbach) = OutOfScope   
      
   or a type theory to give True(S, Goldbach) no content when Goldbach is   
   out of scope, or keep it explicit with an InScope(S, P) family of   
   propositions. Of course, the type theory approach is often easier to use   
   with pencil and paper.   
      
   Is there a conventional alternative to implication for an explicit   
   alternative of type theory?   
      
   Unsatisfying: WhenInScope(S, P) -> (True(S, P) & Foo(P))   
   More satisfying: WhenInScope(S,P,Q in (True(S,Q) & Foo(Q)))   
      
   Hey, I see that in prolog often. Q is an indeterminate (unbound variable   
   in prolog) bound by WhenInScope(S,P,Q in ...) within "...".   
      
   or a lambda expression alternative:   
      
   WhenInScope(S,P,λQ.True(S,Q) & Foo(Q))   
      
   I prefer that over an implicit, semi-ad-hoc type theory.   
      
   Are there conventional names for these ideas and an author and excellent   
   exposition textbook?   
      
   --   
   Tristan Wibberley   
      
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