XPost: comp.theory, sci.logic, sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 29/12/2025 19:53, Richard Damon wrote:   
   > On 12/29/25 2:32 PM, olcott wrote:   
   >> On 12/29/2025 1:21 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.   
   >>>   
   >>   
   >> Unless (as I have been saying for at least a decade)   
   >> the formal language directly encodes all of its   
   >> semantics directly in its syntax. The Montague   
   >> Grammar of natural language semantics is the best   
   >> known example of this.   
   >>   
   >   
   > But it can't, as any system that defines symbols, can have something   
   > outside it assign additional meaning to those symbols.   
      
   Ontology suggests ways to *apply* a system. The system itself works   
   without additional meaning just as it does with. That's the point of   
   formal systems.   
      
   > There may be SOME meaning within the system, but, with a sufficiently   
   > expressive system, additional meaning can be imposed.   
      
   additional meaning is given to an embedding or extension (which is   
   pretty-much a special-case of embedding) of a system, not to the system   
   itself.   
      
   In the case of Gödel's preamble, he defines an extension of PM (I should   
   suppose he was using 2nd ed. in 1931 from his untruths about PM if   
   applied to 1st. ed.) That extension is inconsistent (or, better, I   
   think, indiscriminate). his referent there for PM slides between PM and   
   the derived system as he writes and he gets muddled taking a half-formed   
   conclusion about one, assuming and completing it for the other.   
      
   Then he defines a new system "P" which he uses to get even more muddled,   
   leaves out the crucial elements of his proof because it's too easy to   
   get wrong, and Stephen Meyer says he does get it wrong; he seems to be   
   the only person in the world that ever checked.   
      
   --   
   Tristan Wibberley   
      
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