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| zuhair... |
 Posted: Sun Nov 01, 2009 11:14 am |
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This theory might look like Ackermanns' but yet it differ from
Ackermann in that it doesn't require the second completeness axiom for
V to work in.
Language: FOL with the following primitives:
Identity "="
Epsilon membership "e"
the primitive constant V.
Define: x is a set <-> x e V
Define: x subclass y <-> for all z ( z e x -> z e y )
Axioms:
Extensionality: for all z ( z e x <-> z e y ) ->x=y
Transitive: for all x,y ( x e y & y e V -> x e V )
Class comprehension:
if phi is a formula in which at least y
is free, that do not use V, and in which x is
not free, then all closures of:
Exist x for all y ( y e x <-> ( y e V & phi ) )
are axioms
Define: x={y|phi} <->
for all y ( y e x <-> ( y e V & phi ) )
Define: y transitional of x <->
( x subclass y & y is transitive )
Define: z=Tc(x) <->
( z transitional of x &
for all y ( y transitional of x -> z subclass y ) )
"Tc" stands for "Transitive Closure".
Set comprehension:
if phi is a formula in which at least y or z are
are free, with parameters x1,...xn, that do not
use V, and in which c is not free, then all
closures of
x1 e V & ...& xn e V &
for all c,z ((c= {y|phi} & c subclass Tc(z)) ->~phi(z))
->
Exist a set c for all y ( y e c <-> phi )
are axioms.
Theory definition finished/
For the purposes of this theory, I shall define "nice set" in the
following manner:
x is a nice set <->
( x e V & for all y ( y e Tc(x) -> ~ y e Tc(y) ) )
So there is no circular membership of nice sets.
Now if we restrict ourselves to nice sets then we can easily derive
theorems of pairing,union,power,infinity,separation and Replacement of
nice sets, thus rendering ZF minus Regularity interpreted as a sub-
theory of this theory.
There is no need for the axiom of second completeness for V of
Ackermanns' in this theory, which is
For all x,y ( x subclass y & y e V -> x e V )
This is axiomatized by Ackermann to prove power, but this is not
needed here when working with nice sets.
Zuhair |
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| zuhair... |
| Posted: Sun Nov 01, 2009 12:47 pm |
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On Nov 1, 4:14 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
[quote]This theory might look like Ackermanns' but yet it differ from
Ackermann in that it doesn't require the second completeness axiom for
V to work in.
Language: FOL with the following primitives:
Identity "="
Epsilon membership "e"
the primitive constant V.
Define: x is a set <-> x e V
Define: x subclass y <-> for all z ( z e x -> z e y )
Axioms:
Extensionality: for all z ( z e x <-> z e y ) ->x=y
Transitive: for all x,y ( x e y & y e V -> x e V )
Class comprehension:
if phi is a formula in which at least y
is free, that do not use V, and in which x is
not free, then all closures of:
Exist x for all y ( y e x <-> ( y e V & phi ) )
are axioms
Define: x={y|phi} <-
for all y ( y e x <-> ( y e V & phi ) )
Define: y transitional of x <-
( x subclass y & y is transitive )
Define: z=Tc(x) <-
( z transitional of x &
for all y ( y transitional of x -> z subclass y ) )
"Tc" stands for "Transitive Closure".
Set comprehension:
if phi is a formula in which at least y or z are
are free, with parameters x1,...xn, that do not
use V, and in which c is not free, then all
closures of
x1 e V & ...& xn e V &
for all c,z ((c= {y|phi} & c subclass Tc(z)) ->~phi(z))
-
Exist a set c for all y ( y e c <-> phi )
are axioms.
Theory definition finished/
For the purposes of this theory, I shall define "nice set" in the
following manner:
x is a nice set <-
( x e V & for all y ( y e Tc(x) -> ~ y e Tc(y) ) )
So there is no circular membership of nice sets.
Now if we restrict ourselves to nice sets then we can easily derive
theorems of pairing,union,power,infinity,separation and Replacement of
nice sets, thus rendering ZF minus Regularity interpreted as a sub-
theory of this theory.
There is no need for the axiom of second completeness for V of
Ackermanns' in this theory, which is
For all x,y ( x subclass y & y e V -> x e V )
This is axiomatized by Ackermann to prove power, but this is not
needed here when working with nice sets.
Zuhair
[/quote]
The proof of power in this theory is:
Since x1 is a member of V, and since x1 subclass x1
then Px1V which is the class of all subclasses of x that are members
of V
should contain x1 among its members, now since x1 is
a nice set then then Px1V cannot be a subclass of x1
since by then this would entail that x1 e x1 which is contradictive
because x1 is a nice set! thus Px1 is a set!
Zuhair |
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