XPost: sci.math, comp.theory, "]"   
   From: polcott333@gmail.com   
      
   On 1/21/2026 10:59 PM, Python wrote:   
   > Le 22/01/2026 à 04:54, olcott a écrit :   
   >> 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 actually IS a relationship in the domain of PA. PUNTO.   
   >   
   > It is what it is. Denial is hopeless.   
      
   When PA is actually given its own truth predicate   
   anchored only in its own axioms then for the first   
   time one see that meta-math truth in the standard   
   model of arithmetic never was actually true in PA   
   itself at all.   
      
   --   
   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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