XPost: sci.logic, sci.math, comp.theory   
   From: news.x.richarddamon@xoxy.net   
      
   On 1/18/26 11:28 PM, olcott wrote:   
   > On 1/18/2026 9:56 PM, Richard Damon wrote:   
   >> On 1/18/26 10:19 PM, olcott wrote:   
   >>> On 1/18/2026 7:24 PM, Python wrote:   
   >>>> Le 19/01/2026 à 00:41, olcott a écrit :   
   >>>> ..   
   >>>>> I already just said that the proof and refutation of   
   >>>>> Goldbach are outside the scope of PA axioms.   
   >>>>>   
   >>>>> Any proof or refutation of Goldbach would have to use   
   >>>>> principles stronger than the axioms of PA, because PA   
   >>>>> itself does not currently derive either direction.   
   >>>>   
   >>>> "currently" ? ?  What kind of language is that? PA is what it is, it   
   >>>> not changing with time !   
   >>>>   
   >>>> You could have said that about Fermat's theorem back in the day...   
   >>>> It happens not to be the case.   
   >>>>   
   >>>> You are out of reason, Peter. Not only a liar, an hypocrite, but a   
   >>>> fool.   
   >>>>   
   >>>   
   >>> If its truth value cannot be determined in a finite   
   >>> number of steps then it is not a truth bearer in PA,   
   >>> otherwise it is a truth-bearer in PA with an unknown value.   
   >>>   
   >>   
   >> So, you admit that you don't know how to classify it.   
   >>   
   >> Thus its truth-bearer status is unknown.   
   >>   
   >> Thus, your claim that it is outside of PA is just a LIE.   
   >>   
   >   
   > No it was a mistake. Here is my correction:   
   > If Goldbach's truth value cannot be determined in a   
   > finite number of steps then it is not a truth bearer   
   > in PA, otherwise it is a truth-bearer in PA with an   
   > unknown truth value.   
   >   
   > This has no effect on my claim that I got rid of   
   > Gödel Incompleteness.   
      
   Sure it does. As your system is just not well founded by its own definitios,   
      
   >   
   > When we change the foundation of formal systems   
   > to proof theoretic semantics and add my truth   
   > predicates then Gödel's claim of applying to   
   > every formal system that can do a little bit of   
   > arithmetic becomes simply false.   
      
   But you CAN'T do that and keep the systems.   
      
   >   
   > Every attempt at showing incompleteness  PA   
   > has never actually been  PA.   
      
   Sure it is.   
      
   Godel's G shows your system is not well founded.   
      
   >   
   > The satisfaction of external models of arithmetic   
   > never has been  PA. These are categorically   
   > outside of PA by the definition of proof theoretic   
   > semantics thus defined as non-well-founded. This   
   > neuters their ability to show incompleteness.   
   >   
   >   
      
   But you system is just non-well-founded in PA.   
      
   Godel's G has NO truth value, not even non-well-founded in PA by your   
   system, and thus your system is broken.   
      
   The problem is that for statements like it that have the property of not   
   being having a known truth value if not provable, you system just breaks   
   down.   
      
   There is no proof of it being true, so it can't be true.   
   There is no proof of it being false, so it can't be false.   
   There is no proof of being not-well-founded, so it can't be   
   non-well-founded.   
      
   Your classification of claiming it to be non-well-founded is just   
   non-well-founded.   
      
   In fact, by your systems definitions, the claim of it being   
   non-well-founded is non-well-founded as we can't prove it to be   
   non-well-founded, as if it WAS not-well-founded, that means that you   
   were able to prove that there wasn't a proof of it being false, which   
   means there can't be a number that satisfies the requirement, as any   
   number that existed forms an easy proof of falsehood, and thus must be true.   
      
   So, there CAN'T be a proof of it not being well-founded.   
      
   But if it isn't not-well-founded, then by your definition it must be   
   True or False, which you already said it couldn't be.   
      
   THus the only choice left is it not-well-founded that it is   
   not-well-founded.   
      
   But that arguement extends for that statement, so it is not-well-founded   
   that the not-well-foundedness of the stsatement is not-well-founded.   
      
   Thus, your system breaks with an infinite progression of not being able   
   to classify the truth of the statement.   
      
   So, the reason you think that Godel's (are related) proofs aren't well   
   founded in PA is that your system is just not-well-founded in PA, but   
   refuse to accept it,   
      
   The problem is that definition of Truth is just incompatible with PA,   
   which is why it can't be used.   
      
   The problem is that the system has become "complex" enough that it   
   inherently has grown bigger than provability of all things in it, and   
   thus the concept of Truth being based on Provability just breaks as it   
   means some things have undefinable (not just unknowable) truth values,   
   they can't even be defined as not-having a truth value, as you can't   
   prove that, but you insist that truth must be provable.   
      
   --- SoupGate-Win32 v1.05   
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