XPost: sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 19/01/2026 11:49, Richard Damon wrote:   
      
   ...   
      
   > the concept of Truth being based on Provability just breaks as it   
   > means some things have undefinable (not just unknowable) truth values,   
   > they can't even be defined as not-having a truth value, as you can't   
                        ^^^^^^^   
   I'm pretty sure that's not the right word.   
      
   > prove that, but you insist that truth must be provable.   
      
   Unless you're lucky enough to make a statement about them be an axiom of   
   the system. Then you are hoping you've defined a consistent system but   
   perhaps you got lucky.   
      
   Is it really true, though, that truth based on provability always breaks   
   so? It looks like falsity based on non-provability is the problem and   
   then only in conjunction with some notions of negation and maybe some   
   notions of conjunction too (obviously the Quine might be the problem but   
   we know fixed points give us Quines and vice-versa and they're so   
   important we don't want to lose them).   
      
   What is the negation of "go to the shop" ?   
   What is the negation of "is so! is not! is so! is not! ..." but "is not!   
   is so! is not! is so! ..."   
      
   Given positive intuitionist systems (where a system has unprovable   
   things that are provable in extensions) our truth predicate must leave   
   anything unprovable that could be an axiom of an extension as neither   
   true nor false but rather be inapplicable. A binary Truth predicate (at   
   minimum) is required to even make sense and maybe it requires a further   
   restriction argument (a 2nd order logic, then), which Tarski's   
   indefinability theorem doesn't cover, not by a long way.   
      
   --   
   Tristan Wibberley   
      
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    * Origin: you cannot sedate... all the things you hate (1:229/2)