XPost: comp.theory, sci.math, sci.lang   
   From: chris.m.thomasson.1@gmail.com   
      
   On 11/29/2025 5:10 PM, Richard Damon wrote:   
   > On 11/29/25 7:35 PM, Chris M. Thomasson wrote:   
   >> 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:   
   >   
   > And that was the pre-condition Olcott made of his logic system, that it   
   > have a provability operator.   
   >   
   > Just like you can build a Halt Decider if you assume you have a correct   
   > halt decider (and ignore that it make the system inconsistant).   
      
   With 2-ary, two children per node, root node aside that has four children...   
      
   Parent of nodes 1 and 2 is zero, root.   
      
   Parent of nodes -1 and -2 is zero, root.   
      
   (-2), (-1), (-0+), (+1), (+2)   
      
   This seems rather consistent.?   
      
      
      
      
      
   >   
   >>   
   >> -1     -2   
   >>    \   /   
   >>     \ /   
   >>    (-0+) = the root of all? ;^)   
   >>     / \   
   >>    /   \   
   >> +1     +2   
      
   in 2-ary 0 has the following children (-1, -2, +1, +2), right?   
      
      
      
      
   >>   
   >>   
   >> 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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