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| zuhair... |
 Posted: Mon Nov 02, 2009 2:08 am |
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Guest
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Hi all,
Although I do mostly think that this trial is bout to failure, yet I
will present it, that the main idea behind it might survive to define
a consistent theory in the future.
The basic idea is one that I noticed and wrote here in topic under:
logical set theory. But I will present it in another manner here.
First let have some definitions:
Define: x is {y|phi} <-> For all y ( y e x -> phi(y) )
of course if x is {y|phi} then x is not necessary unique, i.e.
Extensionality do not guaranty the uniqueness of x, with the exception
of course of some predicate that either universal hold for all sets or
the opposite do not hold for any sets.
Define: y transitional of x <-> ( x subclass y & y is transitive )
were "subclass" and "transitive" have their standard meaning.
MST is the set of all sentences entailed( from first order logic with
identity and epsilon membership)by the following non logical axioms:
1. Extensionality: for all z ( z e x <-> z e y ) -> x=y
2. Transitive Closure:
For all x( Exist y ( y transitional of x) &
Exist x' For all z (z e x' <->
for all y (y transitional of x -> z e y))).
From Extensionality x' would be unique and it is said to be the
transitive closure of x. so x' =Tc(x)
3. Comprehension: if phi is a formula in which x is not free, and in
which at least y or z are free, then all closures of
For all x, z (( x is {y|phi} & x subclass z' ) -> ~ phi(z))
OR ~ for all y ( phi(y) -> ~ y e y )
->
Exist x for all y ( y e x <-> phi )
are axioms.
4) Exist x : x e x
Theory definition finished/ |
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