XPost: sci.math, comp.theory, comp.ai.philosophy   
   From: mikko.levanto@iki.fi   
      
   On 18/01/2026 14:53, olcott wrote:   
   > On 1/18/2026 5:18 AM, Mikko wrote:   
   >> 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.   
   >   
   > For nearly a century, discussions of arithmetic have quietly   
   > relied on a fundamental conflation: the idea that   
   > “true in arithmetic” meant “true in the standard model of ℕ.”   
      
   There is no "standard model of N". The symbol "N" is not a name of any   
   theory of arithmetic. It is used as the name of the set of natural   
   numbers or the set of sets that represent the natural numbers.   
      
   THe term "arithmetic" means the same as "the standard model of a theory   
   of arithmetic". The latter expression is rarely used except when talking   
   about some specific theory.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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