XPost: sci.logic, sci.math, comp.theory   
   From: polcott333@gmail.com   
      
   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.”   
   But PA itself has no truth predicate, no internal semantics,   
   and no mechanism for assigning truth values. So what was   
   called “true in arithmetic” was always meta-theoretic truth   
   about arithmetic, imported from an external model and never   
   grounded inside PA.   
      
   This conflation was rarely acknowledged, and it shaped the   
   interpretation of Gödel’s incompleteness theorems, independence   
   results like Goodstein and Paris–Harrington, and the entire   
   discourse around “true but unprovable” statements.   
      
   My work begins by correcting this foundational error with   
   Proof theoretic semantics and non-well-founded is construed   
   as not a truth bearer.   
      
   PA has no internal truth predicate, so classical claims of   
   “true in arithmetic” were always meta-theoretic. My system   
   introduces a truth predicate whose meaning is anchored   
   entirely in PA’s axioms and inference rules, not in external   
   models. Any statement whose meaning requires meta-theoretic   
   interpretation or non-well-founded self-reference is rejected   
   as outside the domain of PA. This yields a coherent, internal   
   notion of truth in arithmetic for the first time.   
      
      
   --   
   Copyright 2026 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)