XPost: comp.theory, sci.logic, sci.math   
   XPost: comp.lang.prolog   
   From: mikko.levanto@iki.fi   
      
   On 16/01/2026 19:47, olcott wrote:   
      
   > The system uses proof-theoretic semantics, where the   
   > meaning of a statement is determined entirely by its   
   > inferential role within a theory. A theory T consists   
   > of a finite set of basic statements together with   
   > everything that can be derived from them using the   
   > inference rules. The statements derivable in this   
   > way are the theorems of T. A statement is true in   
   > T exactly when T proves it.   
      
   Usually the expression "is a theorem of T" is used instead of "is true   
   in T". THe words "true" and "false" are usually reserved for truth in   
   a particular interpretation.   
   > A statement is false   
   > in T exactly when T proves its negation. Some   
   > statements are neither true nor false in T.   
      
   Usually the expression "undecidable in T" is used instead of   
   "neither true nor false in T".   
      
   A theory where some statements that are neither provable or disprovable   
   is said to be incomplere.   
      
   > These are the non-well-founded statements: statements   
   > whose inferential justification cannot be grounded   
   > in a finite, well-founded proof structure. This includes   
   > self-referential constructions such as Gödel-type sentences.   
      
   Gödel's sentence is a sentence of Peano arithmetic. When inpterpreted   
   according to the arithmentic semantics it is not self-referential.   
   Peano's postulates just are insufficient for the proof of Gödel's   
   sentence. The simlest fis is to simply add Gödel's sentence to the   
   postulates. This addition does not create any inconsistency. However,   
   the new theory is still incomlete.   
      
   One should also note that for some theories there is no way to determine   
   whether a claim is provable. A simple example is the group theory.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)