XPost: sci.math, comp.theory, comp.ai.philosophy   
   From: mikko.levanto@iki.fi   
      
   On 15/01/2026 22:37, olcott wrote:   
   > On 1/15/2026 4:02 AM, Mikko wrote:   
   >> On 15/01/2026 07:30, olcott wrote:   
   >>> On 1/14/2026 9:44 PM, Richard Damon wrote:   
   >>>> On 1/14/26 4:36 PM, olcott wrote:   
   >>>>> Interpreting incompleteness as a gap between mathematical truth and   
   >>>>> proof depends on truth-conditional semantics; once this is replaced   
   >>>>> by proof-theoretic semantics a framework not yet sufficiently   
   >>>>> developed at the time of Gödel’s proof the notion of such a gap   
   >>>>> becomes unfounded.   
   >>>>>   
   >>>>   
   >>>> But that isn't what Incompleteness is about, so you are just showing   
   >>>> your ignorance of the meaning of words.   
   >>>>   
   >>>> You can't just "change" the meaning of truth in a system.   
   >>>>   
   >>>   
   >>> Yet that is what happens when you replace the foundational basis   
   >>> from truth-conditional semantics to proof-theoretic semantics.   
   >>   
   >> Gödel constructed a sentence that is correct by the rules of first   
   >> order Peano arithmetic   
   >   
   > within truth conditional semantics and non-well-founded   
   > in proof theoretic semantics. All of PA can be fully   
   > expressed in proof theoretic semantics. Even G can be   
   > expressed, yet rejected as semantically non-well-founded.   
      
   Gödel's sentence is a sentence of Peano arithmetic so its primary   
   meaning is its arithmetic meaning. Peano's postulates fail to   
   capture all of its arithmetic meaning but it is possible to add   
   other postulates without introducing inconsistencies to make   
   Gödel's sentence provable in a stronger theory of natural numbers.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)