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| zuhair... |
 Posted: Sun Nov 01, 2009 1:32 pm |
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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 it.
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, 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 1:44 pm |
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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 x1 that are
members
of V should contain x1 among its members, now any class z which has
Px1V as a subclass of its transitive closure cannot be a subclass of
x1
because if z is a subclass of x1, then the transitive closure of z
would be a subclass of the transitive closure of x1, since Px1V is a
subclass of the transitive closure of z, then Px1V would be a subclass
of the transitive closure of x1, and since x1 e Px1V
then x1 in the transitive closure of x1, thus x1 is not a nice set! a
contradiction
Thus Px1V is not a subclass of x1, thus from the right hand of set
comprehension with phi being "subclass of x1" we get the SET Px1. QED
Zuhair |
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| zuhair... |
| Posted: Sun Nov 01, 2009 3:34 pm |
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Guest
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On Nov 1, 6:32 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 it.
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, 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 x1 that are
members
of V should contain x1 among its members, now any class z which has
Px1V as a subclass of its transitive closure cannot be a subclass of
x1
because if z is a subclass of x1, then the transitive closure of z
would be a subclass of the transitive closure of x1, since Px1V is a
subclass of the transitive closure of z, then Px1V would be a
subclass
of the transitive closure of x1, and since x1 e Px1V
then x1 in the transitive closure of x1, thus x1 is not a nice set! a
contradiction
Thus if any z have Px1V as a subclass of its transitive closure, then
z is not a subclass of x1, thus from the right hand of set
comprehension with phi being "subclass of x1" we get the SET Px1. QED
Zuhair |
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| zuhair... |
| Posted: Sun Nov 01, 2009 5:52 pm |
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[quote]because if z is a subclass of x1, then the transitive closure of z
would be a subclass of the transitive closure of x1,
[/quote]
This needs some clarification:
If z subclass x1, then for any y such that
y transitional of x1, then y transitional of z ( because y
transitional of x1
only mean that y is transitive and x1 subclass of y, now if z is
subclass
of x1 then z would be subclass of y and since y is transitive, then
y
would be transitional of z).
Now the transitive closure of x1 is transitional of x1, thus by the
above reasoning it would be transitional of z also, so we have Tc(x)
transitional of z, so by definition of transitive closure we have Tc
(z) subclass Tc(x), because the transitive closure of any set u is the
subclass of all transitional sets of u.
Zuhair |
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| zuhair... |
| Posted: Mon Nov 02, 2009 12:46 am |
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Guest
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On Nov 1, 6:32 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 it.
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, 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]
I don't know if this theory is consistent or not, perhaps the easiest
way to go about that is to test the equivalence of this theory with
Ackermanns' since both are theories in the same language.
Set comprehension schema seems to be stronger than Reflection schema
of Ackermanns' since it do prove power without the need for the second
completeness axiom for V, something that the reflection schema of
Ackermanns' cannot do.
However I do think that if this theory is consistent then it is most
possibly
equivalent to Ackermanns'.
However this theory contain no ad hoc axiom at all, so its exposition
is rigorous, all axioms contribute nicely to one theme, you can hardly
remove any of its axioms and work independently and get to define any
sets other than empty. Suppose you remove Set comprehension, then the
resulting theory will only define One class that is the empty class,
no way to build other classes; Remove axiom 2 then you cannot define
transitive closures and axiom schema of set comprehension should be
removed, also we end up with one class 0, remove class comprehension
and then we cannot define neither {y|phi} nor can we define transitive
closures, and thus we should remove set comprehension also, resulting
in a theory with V only.
So all axiom interplay tightly into one theme to define classes, from
which we can define sets, then we make classes of sets, then we define
sets of those classes, etc..., in a nice recursive machinery.
Ackermanns' class theory share three axiom schemes with this theory,
namely the first three ones. Only one axiom scheme is different and
most of that scheme also share the same restrictions on formula phi in
it as those in Ackermanns'.
However the second completeness axiom of Ackermanns' is definitely ad
hoc, something that this theory doesn't have.
Foundation is not needed in this theory, the definition of nice sets
is enough for it to work with, I guess that is the same with
Ackermanns'.
However we can add foundation to this theory without any problem. Same
thing to be said of Choice.
All in All, from expositional point of view, this theory is the most
elegant one!
It is an elegant theory of nice sets!
Zuhair |
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