From: matmcinn@nuscc.nus.sg (brett mcinnes)
Subject: Re: TIME HAS INERTIA - GOEDEL's THEOREMS - att: Dr.PRATT, Mc CULLOUGH
Organization: National University of Singapore
X-Newsreader: Tin 1.1 PL4
Date: Mon, 11 Jan 1993 04:01:16 GMT

abian@iastate.edu (Alexander Abian) writes:
: 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
: 
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: -- 
:    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)