system for analyzing set theory

[y.kohmoto]

         [An explaination of A020995 and A097547.]

    Let f(x) denote a function from a class of subsets of a class C to C .

    Consider the following translation between the system {C, f(x)} and ZF
set theory.
    Translation :
              System                   ->            ZF
         Exist B  x el B & f(B)=y      means         x el y
    The system is a good tool for studying set theories.

    Example : The structure of Russel's paradox.
         All x  x el R <-> not(Exist B  x el B & f(B)=x)            -1-
    Use the translation, then you will understand that the right term of the
formula means not(x el x) in ZF.
    Let r denote f(R).
         f(R)=r                                                 -2-
    Input r to x in formula 1.
         x=r
                 r el R <-> not(Exist B  r el B & f(B)=r)            -3-
    If  {r el R} is true, then with formula 2, {r el R & f(R)=r} is true.
    It means, {Exist B  r el B & f(B)=r} is true.                     -4-
    The other hand, the both sides of {<->} must be the same     .
    Hence, {not(Exist B  r el B & f(B)=r} is true.                    -5-
    Formulas 4 and 5  contradict each other.
    So, the assumption that {r el R}is true is not correct.
          {r el R} is false.
    From formula 3, {not( Exist B  r el B & f(B)=r)} is also false.
    Conclusion :
         not (r el R)                                                -6-
         Exist B  r el B & f(B)=r                                    -7-
    Translation :
         not(r el R), so not(r el R & f(R)=r). It means not(Exist B  r el B
& f(B)=r)
         Formula 6 means not(r el r) in ZF.
         Formula 7 means     r el r  in ZF.
    So, mathematicians in ZF observe that the conclusions are a
contradiction.

    They correspond to the last formula of Russel's reasoning.
         All x  x el R <-> not(x el x)
         x=R
                R el R <-> not(R el R)

    We know that this  formula is able to be dissolved with the idea "Set"
in ZF.
    In the system, no additional idea is necessary because both formula 6
and formula 7 are possible to be true.

    If we assume that class C in the system is a finite set which has n
members, then the system becomes a finite model of Free Class.
    The number of finite model of Free Class is the same thing as A097547.
    The number of finite model of Free Class which has natural number is
A020995.

    Yasutoshi