XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   On 12/29/2025 5:06 PM, Richard Damon wrote:   
   > On 12/29/25 4:38 PM, olcott wrote:   
   >> On 12/29/2025 1:53 PM, 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.   
   >>>   
   >>   
   >> "true on the basis of meaning expressed in language"   
   >> can be expressed as relations between finite strings.   
   >   
   > Try to do that.   
   >   
   >   
   >>   
   >>> There may be SOME meaning within the system, but, with a sufficiently   
   >>> expressive system, additional meaning can be imposed.   
   >>>   
   >>> An Montague grammer is out of scope here, as we are talking FORMAL   
   >>> langauges and system, not Natural Language,   
   >>>   
   >>   
   >> "We are therefore confronted with a proposition which   
   >> asserts its own unprovability." (Gödel 1931:39-41)   
   >>   
   >   
   > Right, it is a statement in the meta-theory, commenting on it   
   > unprovabiilty in the base theory.   
   >   
   > Context seems to elude you, because it requires understand.   
   >   
   >   
   >> By using an enormously convoluted process with   
   >> Gödel numbers hiding his actual claim:   
   >   
   >   
   >>   
   >> There exists a sequence of inference steps from   
   >> the axioms of a formal system that prove that   
   >> they themselves do not exist.   
   >   
   > Right, there is an INFININTE string of inference steps in the base   
   > theory that shows that no FINITE string of inference steps to show it.   
   >   
      
   Rene Descartes said: "I think therefore I never existed".   
      
   There is no sequence of inference steps that   
   prove they themselves do not exist.   
      
   There is no sequence of inference steps that   
   prove they themselves do not exist.   
      
   There is no sequence of inference steps that   
   prove they themselves do not exist.   
      
   There is no sequence of inference steps that   
   prove they themselves do not exist.   
      
   There is no sequence of inference steps that   
   prove they themselves do not exist.   
      
   That is all that Gödel ever proved.   
   That is all that Gödel ever proved.   
   That is all that Gödel ever proved.   
   That is all that Gödel ever proved.   
   That is all that Gödel ever proved.   
      
      
   --   
   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)