XPost: comp.theory, sci.math, sci.lang   
   From: chris.m.thomasson.1@gmail.com   
      
   On 11/29/2025 4:17 PM, Richard Damon wrote:   
   > On 11/29/25 6:53 PM, Chris M. Thomasson wrote:   
   >> On 11/29/2025 1:27 PM, Richard Damon wrote:   
   >> [...]   
   >>> Godel proved that such a system can't exist if it can represent the   
   >>> properties of the Natural Number.   
   >   
   > So, where do you have a "provability operator" that will tell you if a   
   > given theory is in fact provable.   
      
   Nope. That is not possible. Think of the integer 0. I can prove that it   
   has, wrt n-ary, n positive children, and n negative children. For   
   example, 2-ary, two (+) and two (-). Say n is a natural number:   
      
   -1     -2   
      \   /   
       \ /   
      (-0+) = the root of all? ;^)   
       / \   
      /   \   
   +1     +2   
      
      
   But that is just for this n-ary case. I cannot just magically   
   extrapolate it our to some programming logic for some random program.   
      
      
      
      
      
   >   
   > That is what he showed can't exist.   
   >   
   > The problem is that there are an infinite number of possible proofs to   
   > see if any of them reach the desired statement.   
   >   
   > You can CHECK if a proof is validly proving the statement, but not   
   > determine if there exist such a proof, as the negative result requires   
   > infinite work.   
      
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