Small Set theory:Revised.
- From: "zuhair" <zaljohar@xxxxxxxxx>
- Date: 16 Dec 2006 10:16:28 -0800
-Small Set Theory-
Primitive e
Definition:
x is P_defined <-> Ay( yex<-> (~y=x & P[y]) )
x is P_embedded<-> Ay( yex<-> (~y=x & P[y]) )&P[x].
x is P_unembedded<-> Ay( yex<-> (~y=x & P[y]) )&~P[x].
Axioms:
Ax.1: Extensionality: AxAy(y=x <-> Az(zex<->zey) )
Ax.2: Comprehension:ExAy( yex<-> (~y=x & P[y]) )
Ax.3: Exclusion:AxAy( (yex & y is P_defined)-> ~P[x] )
Ax.4: Infinity: EN Ax ({}eN /\(xeN->S(x)eN) , were S(x)=xU{x}.
Pluse\minus.
Ax.5: Axiom of choice. As in ZFC.
Theorums.
1) Universe: Ev Ay ( yev ).
2) Pairing:
AaAbExAy( (yex<->(~y=x&(y=a v y=b)))&( (a is P_defined v b is
P_defined) -> ~P[x]) ).
3) Union:
AzExAy((yex<->(~y=x&(Eu:yeu&uez)))&(y is P_defined ->~P[x])).
and x is called the union set of z , i.e. x=Uz.
4) Separation: AzExAy (yex <->(~y=x&(yez&P1(y)))&(y is P_defined
->~P[x]))
5) Replacement: (AuE!y:P(u,y))->AzExAy:yex<->(Euez: P(u,y)&~y=x) & ( y
is P_defined ->~P[x]).
6) Power: AzExAy (yex<->((Au:uey -> uez)&~y=x)& ( y is P_defined ->
~P[x])).
The essence of this theory is that for y to be a member of x it should
fulfill three requirements;
1) y!=x
2) P[y]
3) if P[x] , then y cannot be P_defined.
to explain 3)there cannot be a set z such that P[z] holds, and that
have x as a member were x is P_defined. weather x is P_embedded or x is
P_unembedded.
Example: The set of all set bijectable to 2, here for P <-> bijection
to 2.
if that set is x, then x can't be a member of a set z that has two
elements, since by then z will have P, violating 2.
This set theory has a universe, has emedded sets like the set of all
ordinals, the set of all sets not bijectable to 2, etc....
It doesn't have the restriction of proper classes.
If this theory is consistent it should be something more than ZFC,
because it has all the applications of ZFC, and extra-theorums and
models.
Now I have a question, should I add an axiom stating that for every set
there is a P that defines it.
i.e. should I add Axiom 0) Ax EP (x is P_defined). were P is a formula
in one free variable.
Or there is no need for that?
So the objective of this theory is clear, that is to avoid Russell
paradox without lose of models (N.B. ZFC avoids Russell paradox but
with lose of many models like the set of all sets and the set of all
ordinals, etc.., in reality all of what I call here embedded sets and
sets that containes them as subsets are not in ZFC because they raise
Russell's paradox, while in this theory they exist ).
The main question here: Is this theory consistent?
Zuhair
.
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