XPost: comp.theory, sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 03/01/2026 02:43, André G. Isaak wrote:   
      
   > 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.   
      
      
   Note that "Not a theorem" does not mean exactly the same as "can be   
   shown to be false", generally. Eventually, true and false come down to a   
   mapping to another system, and to another, and to another, all the way   
   down to the appraiser's personal intuition.   
      
   There's some meaning for true and false in episystems that show whether   
   or not a given statement is a theorem of a system that the episystem is   
   being applied to, but then "is a theorem of 𝕾" and "is not a theorem of   
   𝕾" do the job of "is true" and "is false".   
      
   I feel "true" and "false" are such overloaded and carelessly used terms   
   that it's best to avoid them even if you can find an attempt to   
   formalise them. Most apparent formalisations of them are just attempts   
   to connect a real set of formal terms to intuitive notions of truth   
   because discovering truth is the goal of the reader so they demand a   
   connection between formalism and truth beyond the fact that it's true   
   that the formalism is useful. Perhaps one day I'll appreciate some proof   
   system so much I'll use "true" and "false" and then you should tell me   
   to stop it.   
      
      
   --   
   Tristan Wibberley   
      
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    * Origin: you cannot sedate... all the things you hate (1:229/2)