XPost: sci.math, comp.theory, comp.lang.prolog   
   From: mikko.levanto@iki.fi   
      
   On 17/01/2026 17:54, olcott wrote:   
   > On 1/17/2026 3:46 AM, Mikko wrote:   
   >> 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.   
   >   
   > Plain PA has no internal notion of truth; any truth   
   > talk is meta‑theoretic.   
      
   Of course. Truth is a meta-theoretic concept. The corresponding concept   
   about an uninterpreted theory is theorem.   
      
   The statement that there is a sentence that is neither provable nor the   
   negation of a provable sentence does not refer to truth.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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