XPost: comp.theory, sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 02/01/2026 17:24, Mike Terry wrote:   
   > While speaking informally, "theorem" can mean "a mathematical statement   
   > that has a convincing argument for its truth" (e.g. Pythagoras'   
   > theorem), in formal logic "Theorem" and "Theory" have a technical   
   > meaning:  "Theory" being the deductive closure of a set of axioms, and a   
   > Theorem being a sentence of the Theory.    
      
   We should say "theory proper" to account for historical usage of theory   
   for epitheory and theory proper combined. Curry and Feys did that and   
   they were clearly very good at choosing terminology.   
      
      
   > So every Theorem in the Theory   
   > has a "derivation" from the theories axioms.  It is not directly to do   
   > with "truth" in the formal system.  [Of course, we want our system   
   > (including axioms) to be sound, so all Theorems will be true.]   
      
   Ergh, soundness and truth again. One needs a "wrt" in there: against   
   which system (formal or personal-intuitive) will soundness be judged?   
      
   Mainly we want the system to be useful and going further than that is   
   just a matter of degree and of application. By aiming for sound (without   
   mere multiple-world logics to satisfy the need for "wrt") we're really   
   looking for absolute degree and universal application which you're not   
   ever going to demonstrate that you have achieved.   
      
      
   > Also just as an aside, I don't recall that Godel ever talked about   
   > "truth" of his G statement.    
      
      
   Did he talk about any "G" statement at all except in his introductory   
   fallacy where he defines an inconsistent extension of PM to put the   
   reader in mind of the nature of the matter?   
      
   I haven't got through the rest of it yet. Still wondering whether his   
   identity shorthand is as symmetric due to the other axioms of the system   
   as it is in PM, but not having the oomph to slog through it to check   
   (actually, slowly, reading PM1 to understand fully the theory by which   
   PM gets symmetric identity from its asymmetric definitional axiom).   
      
      
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   Tristan Wibberley   
      
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