From: abian@iastate.edu (Alexander Abian)
Subject: TIME HAS INERTIA - GOEDEL's THEOREMS - att: Dr.PRATT, Mc CULLOUGH
Organization: Iowa State University, Ames IA
Date: Sun, 10 Jan 1993 19:54:24 GMT

Summary:  Completeness and two Incomplteness Theorems of Goedel
Keywords: 

TIME HAS INERTIA

STRICTLY FOR DEVOTED READERS OF:  TIME HAS INERTIA

ON GOEDEL's THEOREMS:            att:  Dr. PRATT  and  Mc CULLOUGH


    In our attempt to model THE PLANET EARTH  we construct GLOBES that
can be bought, say, in bookstores.

        Question 1.   DOES A GLOBE CONSTRUCTED ON THE PLANET EARTH AND 
                      LOCATED, say, ON THE NORTH POLE OF THE PLANET
                      EARTH, MODEL THE  PLANET EARTH ?

    This is essentially the central question involved in Goedel's INCOMPLETE-
NESS  theorem.

        The  answer is  NO !

    PROOF.  Assume on the contrary that the Globe that we constructed and
located on the North Pole of the Earth models the Earth.  Then our planet
Earth now has a Globe on its North Pole,  whereas our Globe does not have
a Globe on its North Pole.  Thus, our Globe does not model our Planet
Earth. But this contradicts our assumption.  Thus, our constructed Globe
which is located on the planet Earth does not model the planet Earth.
    
    Hence, our answer "NO"  is proved.

     Note that locating the Globe on the North pole is not essential. 
     Analogous proof can be given no matter where ON THE PLANET EARTH
     the Globe is located.

     Note also that in the above example it is not essential that the Globe 
     is necessarily constructed on the planet Earth.
   
  The essential implication of the above Proof is:


  (G2)    A  MODEL OF  THE PLANET EARTH SYSTEM CANNOT BE PROVED TO
          EXIST ON THE EARTH SYSTEM ITSELF       

   
         Question 2.  DOES A GLOBE OF PLANET EARTH, CONSTRUCTED AND
                      LOCATED ON THE MOON, MODEL THE PLANET EARTH ?
      

         The answer is:  YES !
        
         PROOF.  The construction of the Globe on the Moon and its
locating on the Moon does not alter the Earth.  So it does model the
Earth.          
         
         The essential implication of the above proof is:
     
        
  (E2)    A MODEL OF THE PLANET EARTH SYSTEM CAN BE PROVED TO
          EXISTS ON A MORE POWERFUL EXTENDED  EARTH-MOON SYSTEM


(There are some crackpots who in contradistinction to (G2) claim that
 locating  Globe 1 on the North pole of the Earth, then locating 
 Globe 2 on the North pole of Globe 1, then locating Globe 3 on the
         North pole of Globe 2, then locating Globe 4 on the North pole of
 Globe 3  AND SO ON ... allows the construction of a Model of the system
 on the system itself.  Indeed, these crackpots claim that everyone of 
 the Globes 1, 2, 3, 4, ...  does model the system.  Of course, the
 Professional Guardians of Logic disagree with these crackpots because
 the usage of the words AND SO ON). 

      The (G2) statement above is the heart, the emotional fiber,
 the romantisized, subconsciously powerful and profound version of
 the OFFICIAL AND FORMAL GOEDEL'S SECOND INCOMPLETENESS THEOREM
 which I will mention shortly.  

      First, however, let me mention that by Goedel's CMPLETENESS
Theorem:  
      THE CONSISTENCY OF THE FORMAL FIRST ORDER THEORIES AND
      THE EXISTENCE OF MODELS FOR THEM are equivalent statements, 

i.e.,in a crackpotish way, we can say:
          
   (C1)   A system of statements is consistent if and only if
          there exists a model for that system  

a more crackpotish way, is to say:
   
   (C1)    Consistency is the same thing as having models

where, of course a system of statements is called consistent if no
statement and its negation are both  derivable from the system
(all this in reference to FIRST ORDER THEORIES -consult any
graduate MATH LOGIC or MODEL theoretic textbook- there are
dozens of them)
 
   Now, the OFFICIAL, FORMAL Statement  of Goedel's Second
Incompleteness Theorem (the emotional version of which I gave
in (G2) above)  is:

 (GOEDEL 2)  Let  S  be a formal first order system whose axioms are
             given by some recursive rule. If  S  is consistent and
             the partial recursive functions can be embedded in S, then
             consistency of  S  (i.e., con S) cannot be proved in  S.


    Now, what is Goedel's First Incompleteness Theorem?

   The emotional, romantisized, subconsciously powerful and deep
version of Goedel's First Incompleteness theorem is:

     (G1)  THERE EXISTS NO INFINITE MODEL THAT CANNOT BE EXTENDED
           TO A LARGER MODEL MODELING OBJECTS THAT COULD NOT
           BE MODELED IN THE ORIGINAL MODEL.   

    For instance if the objects of a model are all finite subsets of
integers, we may extend it in variety of ways by various infinite 
subsets of integers.

    The Formal statement of (G1) is:

 (Goedel 1)  Let  S  be a formal first order system whose axioms are 
             given by some recursive rule. If  S  is consistent and the 
             partial recursive functions can be embedded in  S  then 
             there exists an UNDECIDABLE statement p  such that neither  
             p  nor its negation  -p (not  p) can be proved from the 
             axioms of  S.
  
 (for this Formal statement consult any graduate text of  Math Logic
  or Model Theory)

       Clearly, (Goedel 2)  is a special case of (Goedel 1) where  p
can be taken  as "con S".

        Thus,  any formal  system such as  S  will always have an unde-
cidable statement, a fact that shattered into pieces, smashed and
delivered the death blow to  Hilbert's program of trying to prove the
consistency of systems such as  S  (or more powerful than S) by finit-
istic methods.
 
       
       I am exhausted and it is time to listen to Chopin's 24 th etude
in  C  minor.  I think I mistyped in my last nights posting by typing
Chopin's 24 th etude in A minor, I meant Chopin's 23 rd etude in A minor.  

       This morning I had 12 e-mail messages.  As I have repeatedly
mentioned, I do not read  my e-mail messages and all the 12 e-mail 
messages were deleted.  
       
       Please communicate with me  EXCLUSIVELY   via  sci.physics 
       
       Subject; TIME HAS INERTIA.
  
      With love,                       Alexander Abian

















-- 
   The tendency of maintaining the status-quo, Reaction to provocation and
                The tendency of maintaining again a status-quo.  
    TIME HAS INERTIA  and some energy is lost to move Time forward  
  E = mcc  (Einstein)    must be replaced by    E = m(0) exp(-At) (Abian)
Newsgroups: sci.physics,alt.sci.physics.new-theories,sci.skeptic