XPost: sci.math, comp.theory, comp.ai.philosophy   
   From: news.x.richarddamon@xoxy.net   
      
   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.   
      
   And when we look closer we find a bigger problem.   
      
   The problem that you then run into is its truth-bearer status in your   
   proof-theoretics system CAN'T be known to be non-well-founded, as that   
   means that we KNOW we can't find a counter example, and thus the   
   statement must be true. Your proof of no proof of refutation becomes a   
   proof of truth.   
      
   But if it turns out that we can't actually prove it, or refute it, then   
   your system is in trouble, as the value isn't just unknown but doesn't   
   have ANY valid value, as the non-well-founded declearation isn't   
   well-founded.   
      
   Thus, your concept of "truth" when it is tried to be applied to systems   
   like PA becomes internally non-well-founded and internally inconsistant.   
      
   Since Godel proved that there are statements that are true (in a   
   truth-conditional sense) and thus can't be refuted, but also can't be   
   proven even in a truth-conditional system.   
      
   The class of problems that if they can't be proven true or false, MUST   
   have a specific truth value (as the other side is just an specific   
   instance easy to confirm) is fairly common, it says that it is very   
   likely you DO run into the issue in your system for any system that   
   Godel would prove incomplete.   
      
   This is the fundament non-well-foundedness of your idea when it touches   
   system of the level of PA, and why Tarski was able to prove that such   
   systems CAN'T have a Truth Predicate and be consistant.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)