From: abian@iastate.edu (Alexander Abian)
Subject: TIME HAS INERTIA -  LOWENHEIM-SKOLEM PARADOX UNDRESSED
Summary: COUNTING IN A SET-THEORETICAL MODEL
Keywords: ABIAN, UNDRESSES, LOWENHEIM, SKOLEM,PARADOX
                                                 
                                                        11-18-92
Organization: Iowa State University, Ames IA
Date: Wed, 18 Nov 1992 19:11:46 GMT

Dear Messrs  PRATT , BUDNIK et al,

      I will construct two examples of Set-Theories  C and U  such that 
in C the set {1,2} is countable and in U the set {1,2}  is uncountable!


      As usual,  {a,b,...}  will stand for a set whose elements are  a, b,...
and the set  {{a}, {a,b}}  will be denoted by  (a,b)  (ordered pair).  So,
the elements of the set (a,b)  are   {a}  and  {a,b}.
 
     
   The sets of my set-theoretical model  C  are given below:
 
 
{ }   denoted by  0,      {0}   denoted by  1,      {0,1}  denoted by 2

 {{0},{0,1}}  which is same as  {1,2},      {1},   {{1},{1,2}}    and
 
{{{0},{0,1}}, {{1},{1,2}}}    denoted by  {(0,1), (1,2)} = F  
  
   
So, the entire set-theoretical model C  has  the ABOVE SETS - NOTHING MORE.
Of course, it is not a model for ZF set-theory.  However, it is a model
for some set-theory.  The union of two sets of the Model may or may not
exist in the Model.  The intersection of two sets may or may not exist, 
the powerset of a set  may or may not exist.  The Cartesian product of 
two sets may or may not  exist.  The singleton of a set may or may not exist
in the Model, etc., etc  It is a very, very, very weak set-theoretical model
BUT IT IS A SET-THEORETICAL MODEL ! 

 
 THEOREM.  In model C  the set  {1,2}  is COUNTABLE.
 
 PROOF.  Because in  C  there exists the Function  F = {(0,1), (1,2)}  
         which establishes a one-to-one correspondence between  the na-
         tural number  2  and the set {1,2}  (and every mentioned set in the
         proof is a set of  C).  But this means that  {1,2}  is
         COUNTABLE IN Model C (in fact, countable by the natural number 2)

     
  Next, I introduce my set-theoretical model U  which has the same sets as
  model  C, except that it does not have the set  {{{0},{0,1}}, {{1},{1,2}}}

    
    Using notations introduced in  C, the sets of my set-theoretical model 
 U are given below: 
 
         { } ,  {0},  {0,{0}},   {1},   {0,1},   {1, 2} 
 
    Again,  C  is a very, very, very, very weak  set-theoretical model BUT IT
IS A SET-THEORETICAL MODEL!

 
    THEOREM.  In model  U  the set  {1,2}  is UNCOUNTABLE.

 PROOF.  Because in  U  there is no function  F  which establishes a 
one-to-one correspondence between a natural number and  the set  {1,2}. 


    P.S. This is the undressing of the LOW-SKO. paradox.  I have also
an undressing proof for the LOW-SKOLEM THEOREM.  But I have no time
to undress so many people.  However, I may undress two more things:
" Derivation of addition of velocities in Relativity"
and  " Derivation of Various non-Euclidean Geometries". 
But maybe I won't do these  - it may be too provocative !   Remember
(A1) ! - reaction to provocation  and  H-bomb !! 
  
    Withe best wishes and regards,

                                           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