XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   On 12/29/2025 2:20 PM, Tristan Wibberley wrote:   
   > 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.   
   >   
      
   Gödel proved that there cannot possibly exist any   
   sequence of inference steps in F prove that they   
   themselves do not exist.   
      
   He admitted this himself:   
   ...We are therefore confronted with a proposition   
   which asserts its own unprovability. 15 … (Gödel 1931:40-41)   
      
   Gödel, Kurt 1931.   
   On Formally Undecidable Propositions of   
   Principia Mathematica And Related Systems   
      
      
   --   
   Copyright 2025 Olcott

   
      
   My 28 year goal has been to make 
   
   "true on the basis of meaning expressed in language"
   
   reliably computable.

   
      
   This required establishing a new foundation
   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)