XPost: comp.theory, sci.math, sci.lang   
   From: chris.m.thomasson.1@gmail.com   
      
   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.   
      
   I hope this can exist. Sorry for any typos with the n-ary tree, n=2   
   here. Can you notice any errors I missed?  natural number in the tree   
   has two unique children. I can derive these children from any natural   
   number. I can get at a child's parent just from its mapped natural. It's   
   100% full circle.   
      
          0   
         / \   
        /   \   
       1     2   
      / \   / \   
     3   4 5   6   
   ...........   
      
   The children of 1 are:   
      
   c[0] = 1 * 2 + 1 = 3   
   c[1] = c[0] + 1  = 4   
      
      
   Nice! Now, to map back   
      
   The parent of 3 is:   
      
   p = ceil(3 / 2) - 1 = 1   
      
      
   The parent of 4 is:   
      
   p = 4 / 2 - 1 = 1   
      
      
   The parent of 5 is:   
      
   p = ceil(5 / 2) - 1 = 2   
      
   The parent of 6 is:   
      
   p = 6 / 2 - 1 = 2   
      
      
   Notice I do not have to use ceil in the case of 2-ary when the natural   
   number in question is even? Premature optimization? ;^)   
      
   It works even with using ceil all the time:   
      
      
   Take the parent of 3 and 4:   
      
   p = ceil(3 / 2) - 1 = 1   
   p = ceil(4 / 2) - 1 = 1   
      
      
   Lets try a parent at zero with its 2-ary children of 1 and 2:   
      
   p = ceil(1 / 2) - 1 = 0   
   p = ceil(2 / 2) - 1 = 0   
      
      
   ^D   
      
      
   I need to adapt it for negative numbers. Think of the following 2-ary tree:   
      
      
   -1   -2   
      \ /   
       0   
      / \   
   +1   +2   
      
      
   So, lets try it out... The children on the negative side of zero. Flip   
   things wrt +1 becomes -1:   
      
   c[0] = 0 * 2 - 1 = -1   
   c[1] = c[0] - 1  = -2   
      
   Well, that works! Let's get the parent of -2, and flip the sign on the   
   -1 to +1, should be zero: Also, lets flip ceil to floor:   
      
   p = floor(-2 / 2) + 1 = 0   
      
   Nice, lets try -1:   
      
   p = floor(-1 / 2) + 1 = 0   
      
      
   It works... Interesting to me.   
      
   Lets try -3, its parent should be -1:   
      
   p = floor(-3 / 2) + 1 = -1   
      
   Also, -4's parent should be -1:   
      
   p = floor(-4 / 2) + 1 = -1   
      
   Nice!   
      
   -5 and -6 should both have a parent of -2:   
      
      
   p = floor(-5 / 2) + 1 = -2   
   p = floor(-6 / 2) + 1 = -2   
      
   perfect.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)