[continued from previous message]   
      
   >>>>> After all, General Knowledge is an inconsistent set of information   
   >>>>>   
   >>>>   
   >>>> You just can't comprehend that knowledge is   
   >>>> structured in an acyclic directed graph.   
   >>>   
   >>> Nope, it is cyclic,   
   >>   
   >> Show a concrete example of knowledge itself being cyclic.   
   >   
   > Try to define ANY word, and the words used to define it, and so one till   
   > you get to a word that just is without a defintion.   
   >   
   > You will always eventually cycle back to a word you have already used.   
   >   
      
   A concrete example is a specific word that does this.   
      
   >>   
   >>>  as our base facts of knowledge are interrelated. The is on one root   
   >>> fact.   
   >>>   
   >>   
   >> You have a type that makes your sentence gibberish.   
   >   
   > Yes, I have a typ*o*, as did you   
   >   
   > There is no one root fact in our knowledge.   
      
   {Thing} is the root of the knowledge ontology.   
      
   > If every fact has other   
   > facts that it is based on, there is no root fact, and the system, since   
   > it is finite, is cyclical.   
   >>   
   >>>>   
   >>>>>>   
   >>>>>>> It can not handle most systems with a countably infinite domain   
   >>>>>>> of regard, so not Natural Numbers, not Finite Strings, not Turing   
   >>>>>>> Complete Systems.   
   >>>>>>>   
   >>>>>>   
   >>>>>> It can handle them at least to the same extent   
   >>>>>> as humans minds. Algorithmic compression.   
   >>>>>   
   >>>>> NOPE, As if it could handle Natural Numbers, then we could create   
   >>>>> the G for the system, and it couldn't prove it.   
   >>>>>   
   >>>>   
   >>>> It is merely that diagonalization hides the semantic   
   >>>> incoherence that reject's G.   
   >>>   
   >>> Nope. It seems you don't understand that G is just a statement that   
   >>> no number statisfies a specific (complicated) Primitive Recursive   
   >>> Relationship. A Relationship that can ALWAYS, for ANY number, be   
   >>> evaluated in finite time.   
   >>>   
   >>   
   >> "that no number satisfies a specific (complicated) Primitive   
   >> Recursive Relationship" How is this shown?   
   >   
   > In the meta-theory that undrstands the added meaning to the numbers.   
   >   
      
   Is a separate thing thus not:   
      
   A system such all semantic meaning of the formal   
   system is directly encoded in the syntax of the   
   formal language of the formal system making   
   ∀x ∈ L (Provable(L,x) ≡ True(L,x))   
      
   > In the base theory, the numbers do not have that meaning, but, since   
   > this meta-theory developes a method to express as a single number ANY   
   > statement/collection of statements that can be expressed in the theory,   
   > and, because of the structure of these numbers, can check if a given the   
   > given statement is a proof for another statement. And the PRR is an   
   > embodeyment of such an algorithm to check if a statement is a proof of   
   > the statement G.   
   >   
      
   You merely ignored my specified requirements. (see above).   
   It is possible to defined f-cked up systems that are incomplete   
   That is not the same thing as all systems are necessarily incomplete.   
      
   > THus *ANY* proof of G, will create a number which will satisfy that PRR,   
   > thus making G false.   
   >   
   > Since it is impossible for there to be a correct proof of a false   
   > statement, there can not be a number that satisfies the PRR of G, as   
   > that would lead to the contradiction.   
   >   
   > This means that no number can staisfy that PRR, and thus G must be true.   
   >   
   > This proof can only be done in the meta-system that has the knowledge of   
   > the meaning of all the numbers, and there are an infinite number of   
   > possible systems assigning meaning, so we can't just search them all.   
   >   
   > The key is that the meta-system was specifically constructed so that   
   > statements, like that G, and its PRR, that do not reference the   
   > additional "facts" the create the meaning, will have the same truth   
   > values in the two systems.   
   >   
   > Thus, since G and the PRR meet that requirement, our knowledge in the   
   > Meta-System transfers to the base system, but the proof, that uses that   
   > knowledge does not.   
   >   
   >>   
   >>> There is no "diagonalization" in G. You are confusing different proof.   
   >>>   
   >>> The question of G is a pure mathematical question, either a number   
   >>> does or does not satisfy it.   
   >>>   
   >>> In other words, your "logic" says some questions with factual answers   
   >>> are just wrong.   
   >>>   
   >>> In other words, your logic is proven to be self-inconsistant, as   
   >>> statements provably true are considered to be illogical.   
   >>>   
   >>   
   >> You know that the Liar Paradox: "This sentence is not true"   
   >> is not a truth bearer. None-the-less when we add one level   
   >> of indirect reference   
   >> This sentence is not true: "This sentence is not true"   
   >> it becomes true.   
   >>   
   >   
   > Yes. So?   
   >   
   > The statement G has a truth value, because no number does satisfy the   
   > PRR, and thus it is true.   
   >   
   > It can not be proven in the base system, as the only verification would   
   > require testing EVERY finite value, of which there are an infinite   
   > number of them, so it doesn't form a proof, but does establish its truth.   
   >   
   > This is not true of the Liar's paradox. It just can not have a truth value.   
   >   
      
   It does not have a truth value in the theory:   
   "This sentence is not true"   
   because of pathological self-reference   
      
   It does have a truth value in the meta-theory:   
   This sentence is not true: "This sentence is not true"   
   because of pathological self-reference has been eliminated.   
      
   > This comes from the fundamental difference between the statements of "I   
   > am not True" and "I am not Provable".   
   >   
   > The first can not have a truth value, as either value creates a   
   > contradiction.   
   >   
   > This is not true of the second. It can not be false, as if it is false,   
   > then it is provable, so it must be true (as we can only prove true   
   > statements in a non-contradictory system),   
   >   
   > It CAN be True, as there is no actual requirement of True statments   
   > being provable, as Truth can come out of an infinite number of steps of   
   > implication.   
   >   
   > It also can be a non-truth-bearer if there isn't anything that makes it   
   > true.   
   >   
   > The key to the proof is that the statement isn't just a statement of   
   > that form, but a statement that must have a truth value as it is a   
   > statement is about something which follows the law of the excluded   
   > middle, that only derives that meaning when we add additional   
   > information from the meta-system.   
      
      
   --   
   Copyright 2025 Olcott

   
      
   My 28 year goal has been to make 
   
   "true on the basis of meaning expressed in language"
   
   reliably computable.

   
      
   This required establishing a new foundation
   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)