XPost: sci.logic, sci.math, comp.theory   
   From: news.x.richarddamon@xoxy.net   
      
   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.   
      
   And that R(x) is just a basic mathematical realtionship that is fully   
   defined in PA.   
      
   This particular R is fairly conplex, but that is just because it has a   
   number of terms in it, but it basicaaly of the same type as the isPrime   
   relationship which can be defined as:   
      
   P(x) is true if x is a natural number greater than 1 and there is no   
   pair of natural numbers between 1 and x-1, a and b, such that a * b = x.   
      
   R(x) is exactly of the same type of statement as this, all qualification   
   are over finite sized sets, and thus all decisions are effectively   
   computable.   
      
   IF you are going to use the interpreation of G in the meta, then you   
   need to accept that the interpretation is valid which just breaks you   
   proof, as you need to look at the CORRECT and PRECISE interpreation, and   
   not just the natual language version of the interpreation.   
      
   >   
   > 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.   
      
   Since you don't start from the actual G_F, you claim is just unsound.   
      
   >   
   > 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.   
   >   
      
   But there isn't a self-refernce to block.   
      
   And then you have the problem that proof-thoretical interpreations is   
   not-well-founded in PA, because it CAN'T assign a "truth value" to   
   statements like the actual G. (that there doesn't exist a number that   
   statisfies a relationship) as this class of statement just can not be   
   not-well-founded, as ANY proof that tries to show the statement being   
   not-well-founded turns out to actually establish the truth value of the   
   statement, as to prove that you can't prove the that the statement can't   
   be false means proving that no such number exists, which proves the   
   statement to be true.   
      
   This is irregardless of claims of unprovability. Statements of   
   qualification over a set with a total computable function just can't   
   ever be non-well-founded. As proving that the side that only needs one   
   number can't exist, proves the other side that no such number exists.   
      
   >   
   >>>   
   >>>> 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.   
   >>>   
   >>>   
   >>   
   >   
   >   
      
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