XPost: comp.theory   
   From: pa@see.signature.invalid   
      
   In sci.logic Richard Damon  wrote:   
   > On 12/29/25 2:21 PM, Pierre Asselin wrote:   
   > > In sci.logic Tristan Wibberley  wrote:   
   > >> On 29/12/2025 13:37, Richard Damon wrote:   
   > >   
   > >>> Incompleteness is a property of a given Formal System, it says that   
   > >>> there exist a statement that is true in that system, but can not be   
   > >>> proven in that system.   
   > >   
   > >> What do you mean by "proven" here. Do you mean "derived" ?   
   > >   
   > > I think Richard misspoke slightly. The undecidable statement is   
   > > true *in the intended interpretation* of the formal system   
   > > (In Goedel's case, the natural numbers with addition and multiplication).   
   > >   
   > > Truth "in the formal system" isn't really defined. You need an   
   > > interpretation.   
   > >   
      
   By the way when I wrote "Richard misspoke slightly" I should have   
   added "but that doesn't invalidate his argument". Sorry about that.   
      
   > No, statements in a formal system are DEFINED to be true, if that   
   > statement, referencing object defined in the system model, and related   
   > by relationships defined in the system  can be established starting with   
   > the initial "facts" (axioms) of the system, and following the allowed   
   > logical operations of the system.   
      
   That's provability, not truth.   
      
   > THus in the formal system of addition of Natural Numbers, the statement   
   > 2 + 3 = 5 is a true statement, as it can be derived from the operations   
   > in the system.   
      
   A statement provable in the system, and a true statement about natural   
   numbers.   
      
   > Some Formal System include a "model" that define   
   > interpreations, though another layer above can be added (as Godel did).   
      
   I dunno, I always saw the models as separate from the formal system.   
   (That said, an intended model often provides the motivation for the   
   formal system.)   
      
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   pa at panix dot com   
      
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