XPost: sci.math, comp.theory, comp.lang.prolog   
   From: polcott333@gmail.com   
      
   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. To work proof‑theoretically,   
   we must add a rule‑anchored truth predicate in the   
   sense of Curry, governed by elementary theorems of T.   
   If we then impose an object‑level well‑foundedness   
   constraint on truth—rejecting any cyclic truth   
   dependencies—Gödel’s fixed‑point sentence G becomes   
   syntactically non‑well‑founded and is blocked before   
   any truth value is assigned. In such a system,   
   Gödel’s G is not a deep undecidable truth, but   
   an ill‑formed attempt at self‑reference.   
      
      
   --   
   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)