XPost: sci.logic, sci.math, comp.theory   
   From: polcott333@gmail.com   
      
   On 1/21/2026 6:35 AM, Richard Damon wrote:   
   > On 1/20/26 11:54 PM, olcott wrote:   
   >> On 1/20/2026 10:04 PM, Richard Damon wrote:   
   >>> On 1/20/26 4:23 PM, olcott wrote:   
   >>>> On 1/19/2026 11:29 PM, Richard Damon wrote:   
   >>>>>> My system is not supposed to decide in advance whether   
   >>>>>> Goldbach is well‑founded. A formula becomes a truth‑bearer   
   >>>>>> only when PA can classify it in finitely many steps.   
   >>>>>> Goldbach may or may not be classifiable; that’s an open   
   >>>>>> computational fact, not a semantic requirement. This has   
   >>>>>> no effect on Gödel, because Gödel’s sentence is structurally   
   >>>>>> non‑truth‑bearing, not merely unclassified.   
   >>>>>   
   >>>>> Which shows that you don't understand what logic systems are.   
   >>>>>   
   >>>>> The don't "Decide" on truths, they DETERMINE what is true.   
   >>>>>   
   >>>>> Your problem is that either there is, or there isn't a finite   
   >>>>> length proof of the statement.   
   >>>>>   
   >>>>> Semantics can't change in a formal system, or they aren't really   
   >>>>> semantics.   
   >>>>>   
   >>>>> Your problem is you don't understand Godel statement, as it *IS*   
   >>>>> truth bearing as it is a simple statement with no middle ground,   
   >>>>> does a number exist that satisfies a given relationship. Either   
   >>>>> there is, or there isn't. No other possiblity.   
   >>>>>   
   >>>>> You confuse yourself by forgetting that words have actual meaning,   
   >>>>> and that meaning can depend on using the right context.   
   >>>>>   
   >>>>> Godel's G is a statement in the system PA.   
   >>>>>   
   >>>>> It is a statement about the non-existance of a natural number that   
   >>>>> satisfies a particular computable realtionship.   
   >>>>>   
   >>>>> It is a statement defined purely by mathematics and thus doesn't   
   >>>>> "depend" on other meaning.   
   >>>>>   
   >>>>> It is a mathematical FACT, that for this relationship, no matter   
   >>>>> what natural number we test, none will satisfy it, so its   
   >>>>> assertation that no number satisfies it makes it true.   
   >>>>   
   >>>> PA augmented with its own True(PA,x) and False(PA,x)   
   >>>> is a decider for Domain of every expression grounded   
   >>>> in the axioms of PA.   
   >>>   
   >>> No, it becomes inconsistant.   
   >>>   
   >>>>   
   >>>> A system at a higher level of inference than PA can   
   >>>> reject any expressions that define a cycle in the   
   >>>> directed graph of the evaluation sequence of PA   
   >>>> expressions. Then PA could test back chained inference   
   >>>> from expression x and ~x to the axioms of PA.   
   >>>>   
   >>>   
   >>> But there is no "cycle" in the statement of G. It is PURELY a   
   >>> statement of the non-existance of a number that satisfies a purely   
   >>> mathematic relationship (which has no meaning by itself in PA).   
   >>>   
   >>   
   >> Even the relationship cannot exist  PA.   
   >> Instead it is about PA in outside model theory   
   >   
   > No, it doesn't mention PA, it is about the numbers that are IN PA.   
   >   
   > Your problem is you forget to actually know what Godel's G is, a you   
   > only read the Reader's Digest version of the proof, as that is all you   
   > can understand.   
   >   
   > That, or you are saying that mathematics itself isn't in PA, and that   
   > you proof-theoretic stuff isn't in PA either,   
   >   
   > Sorry, you are just showing how ignorant you are.   
   >   
      
   G_F ↔ ¬Prove_F(Gödel_Number(G_F)) contains a semantic   
   dependency loop, because evaluating G_F requires   
   evaluating Prove_F on the Gödel number of G_F, which   
   in turn requires evaluating G_F again;   
      
   this cycle in the directed graph of its evaluation   
   sequence makes the formula non‑well‑founded at the   
   meta‑mathematical level, and under a well‑founded   
   proof‑theoretic semantics such expressions are   
   filtered out before interpretation, so the diagonal   
   sentence never enters PA at all.   
      
   In that framework Gödel’s incompleteness construction   
   never gets off the ground—not because Gödel erred, but   
   because the well‑foundedness criterion he didn’t have   
   in 1931 blocks the self‑referential step that his proof   
   relies on.   
      
      
   >>   
   >>> You only can find a cycle when you accept the interpretations in the   
   >>> meta-math.   
   >>>   
   >>> So, do you accept that interpreation (and thus the proof) or do you   
   >>> reject it, and thus have no grounds to deny the effect of the proof.   
   >>   
   >>   
   >   
      
      
   --   
   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)