XPost: sci.logic, sci.math, comp.theory   
   From: polcott333@gmail.com   
      
   On 1/21/2026 9:37 PM, Richard Damon wrote:   
   > On 1/21/26 10:45 AM, olcott wrote:   
   >> 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;   
   >   
   > But that isn't G_F   
   >   
   > G_F is a statement that a particular relationship (lets call it R(x) )   
   > will not be satisfied for any natural number x.   
   >   
      
   That relationship has never existed inside actual   
   arithmetic; it exists only in meta‑mathematics, which   
   people often misconstrue as arithmetic itself.   
   Satisfaction is not a notion available within   
   arithmetic—only within models of arithmetic.   
      
      
   --   
   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   
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