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| zuhair... |
 Posted: Sat Oct 31, 2009 3:21 pm |
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Guest
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I call this theory "logical set theory" simply because the main
motivation behind it is "logical" in the sence that this theory
follows a logical approach to avoid paradoxes.
LST is the set of all sentences entailed (from FOL with identity,
membership and the primitive constant V) by the following non logical
axioms.
1) Extensionality: For all z ( z e x <-> z e y ) -> x=y
2) Class Comprehension:if Phi is a formula 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.
3) Set Comprehension: IF Phi(y,x1,...,xn) is a formula which do not
use V, and in which y,x1,...,xn are its sole free variables, and in
which s is not free, and if Q1,...,Qm are all the sub-formulas of Phi
in which y or c1 or...or cm are free, having no parameters other than
those in Phi, then
For all c1,...,cm& x1 e V,...,xn e V (
~(Q1(c1) & c1={y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm(cm) & cm={y|Qm}& For all y ( Qm -> ~yey ))
-> Exist a set s for all y ( y e s <-> Phi(y,x1,...,xn ) ).
are axioms.
4) Anti-foundation: Exist x: x e x
5) Transitive: For all x , y ( y e x & x e V -> y e V ).
Theory definition finished/
Zuhair
It is interesting to see that even Leśniewski's paradox is not a
paradox in this theory:
Leśniewski's paradox is about the set of all singletons that are not
members of their sole members, now a singleton set containing this set
would be clearly paradoxical.
The set of all singleton's do exist in this theory definitely, but yet
it doesn't lead to any paradox, since the singleton axiom is not a
theorem in this theory, so the set of all singleton's simply do not
have singleton set that contains it as its sole member. However this
set is of course not well founded since it can contain
members like {{U}} were U is the set of all sets.
The rules of pairing, union, power, seem to be restricted to well
founded sets, or at most to sets that permit no circular membership
(set in which all members of their transitive closure are not in their
transitive closure).
Infinity is a theorem here also.
Since we cannot use formulas like ~yey, y is ordinal, y is well
founded, etc..., since power is defined for well founded sets, then it
is obvious that Russell's, Burali-Forti, and Cantor's paradox are all
avoided.
It is also interesting to see that the set of all finite sets in this
theory is a set!
However this theory turns to be very complex when go out of well
founded sets, and it is very difficult to set forth any general rules
in such a harsh grounds.
Zuhair |
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| hagman... |
| Posted: Sun Nov 01, 2009 2:53 am |
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Guest
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On 1 Nov., 02:21, zuhair <zaljo... at (no spam) yahoo.com> wrote:
[quote]I call this theory "logical set theory" simply because the main
motivation behind it is "logical" in the sence that this theory
follows a logical approach to avoid paradoxes.
LST is the set of all sentences entailed (from FOL with identity,
membership and the primitive constant V) by the following non logical
axioms.
1) Extensionality: For all z ( z e x <-> z e y ) -> x=y
2) Class Comprehension:if Phi is a formula 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.
3) Set Comprehension: IF Phi(y,x1,...,xn) is a formula which do not
use V, and in which y,x1,...,xn are its sole free variables, and in
which s is not free, and if Q1,...,Qm are all the sub-formulas of Phi
in which y or c1 or...or cm are free, having no parameters other than
those in Phi, then
For all c1,...,cm& x1 e V,...,xn e V (
~(Q1(c1) & c1={y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm(cm) & cm={y|Qm}& For all y ( Qm -> ~yey ))
-> Exist a set s for all y ( y e s <-> Phi(y,x1,...,xn ) ).
[/quote]
I assume "s is a set" is the same as "s e V" throughout ?
[quote]
are axioms.
4) Anti-foundation: Exist x: x e x
5) Transitive: For all x , y ( y e x & x e V -> y e V ).
Theory definition finished/
Zuhair
It is interesting to see that even Leśniewski's paradox is not a
paradox in this theory:
Leśniewski's paradox is about the set of all singletons that are not
members of their sole members, now a singleton set containing this set
would be clearly paradoxical.
The set of all singleton's do exist in this theory definitely,
[/quote]
Hm, how do you see this?
Probably by 3) with
Phi(x):<=> E y: A z: ( yex & ( zex -> z=y ))
I'm not completely sure what is meant with "all the subformulas of
Phi"
Maybe
Q1(x,y):<=> A z: ( yex & ( zex -> z=y ))
Q2(x,y,z):<=> ( yex & ( zex -> z=y ))
and so on or maybe just the "first level" Q1.
Anyway, it appears that the Qi should only have one free variable(?)
so there seems to be no applicable subformula, i.e. m=n=0, hence
the corresponsing instance of 3) is actually
Exist a set s for all y ( y e s <-> Phi(y) ).
Is this the correct reasoning or could you elaborate more on
the subformulas?
[quote]but yet
it doesn't lead to any paradox, since the singleton axiom is not a
theorem in this theory,
[/quote]
Do you mean that
A xeV: E seV: A y: (yes <-> y=x)
is not a theorem?
I'd try with Phi(y,x1):<-> y=x1
In my eyes, Phi has no subformulas at all (m=0, n=1), thus 3) gives
A x1 e V: E seV: Ay: (yes <-> y=x1)
[quote]so the set of all singleton's simply do not
have singleton set that contains it as its sole member. However this
set is of course not well founded since it can contain
members like {{U}} were U is the set of all sets.
[/quote]
If U is the set of all sets, what is the intended interpretation of V?
[quote]
The rules of pairing, union, power, seem to be restricted to well
founded sets, or at most to sets that permit no circular membership
(set in which all members of their transitive closure are not in their
transitive closure).
Infinity is a theorem here also.
Since we cannot use formulas like ~yey, y is ordinal, y is well
founded, etc..., since power is defined for well founded sets, then it
is obvious that Russell's, Burali-Forti, and Cantor's paradox are all
avoided.
It is also interesting to see that the set of all finite sets in this
theory is a set!
However this theory turns to be very complex when go out of well
founded sets, and it is very difficult to set forth any general rules
in such a harsh grounds.
Zuhair[/quote] |
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| zuhair... |
| Posted: Sun Nov 01, 2009 4:54 am |
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Guest
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On Nov 1, 7:53 am, hagman <goo... at (no spam) von-eitzen.de> wrote:
[quote]On 1 Nov., 02:21, zuhair <zaljo... at (no spam) yahoo.com> wrote:
I call this theory "logical set theory" simply because the main
motivation behind it is "logical" in the sence that this theory
follows a logical approach to avoid paradoxes.
LST is the set of all sentences entailed (from FOL with identity,
membership and the primitive constant V) by the following non logical
axioms.
1) Extensionality: For all z ( z e x <-> z e y ) -> x=y
2) Class Comprehension:if Phi is a formula 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.
3) Set Comprehension: IF Phi(y,x1,...,xn) is a formula which do not
use V, and in which y,x1,...,xn are its sole free variables, and in
which s is not free, and if Q1,...,Qm are all the sub-formulas of Phi
in which y or c1 or...or cm are free, having no parameters other than
those in Phi, then
For all c1,...,cm& x1 e V,...,xn e V (
~(Q1(c1) & c1={y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm(cm) & cm={y|Qm}& For all y ( Qm -> ~yey ))
-> Exist a set s for all y ( y e s <-> Phi(y,x1,...,xn ) ).
I assume "s is a set" is the same as "s e V" throughout ?
are axioms.
4) Anti-foundation: Exist x: x e x
5) Transitive: For all x , y ( y e x & x e V -> y e V ).
Theory definition finished/
Zuhair
It is interesting to see that even Leśniewski's paradox is not a
paradox in this theory:
Leśniewski's paradox is about the set of all singletons that are not
members of their sole members, now a singleton set containing this set
would be clearly paradoxical.
The set of all singleton's do exist in this theory definitely,
Hm, how do you see this?
Probably by 3) with
Phi(x):<=> E y: A z: ( yex & ( zex -> z=y ))
I'm not completely sure what is meant with "all the subformulas of
Phi"
Maybe
Q1(x,y):<=> A z: ( yex & ( zex -> z=y ))
Q2(x,y,z):<=> ( yex & ( zex -> z=y ))
[/quote]
Ok let me examine these formulas:
first lets take Q1 which is E y: A z: ( yex & ( zex -> z=y ))
now this is a formula with one parameter which is y while the
principal free variable is x.
Now for any Qi to be a subformula of Q1, then it should have x as the
principal free variable and if it has a parameter than it should be y
only.
Lets examine Q2, we see that z is a parameter, which is not a
parameter in Q1, thus Q2 doesn't qualify as a sub-formula of Q1.
[quote]and so on or maybe just the "first level" Q1.
Anyway, it appears that the Qi should only have one free variable(?)
[/quote]
Why??? Q1 itself is a sub-formula of Q1 and it has two free variables.
[quote]so there seems to be no applicable sub-formula,
[/quote]
wrong: Q1 is a sub-formula of itself.
i.e. m=n=0, hence
what is m and what is n?
[quote]the corresponsing instance of 3) is actually
Exist a set s for all y ( y e s <-> Phi(y) ).
Is this the correct reasoning or could you elaborate more on
the subformulas?
[/quote]
No its not correct at all.
[quote]
but yet
it doesn't lead to any paradox, since the singleton axiom is not a
theorem in this theory,
Do you mean that
A xeV: E seV: A y: (yes <-> y=x)
is not a theorem?
[/quote]
Yes.
[quote]I'd try with Phi(y,x1):<-> y=x1
In my eyes, Phi has no subformulas at all (m=0, n=1),
[/quote]
Same error above, Phi itself is a subformula of itself.
thus 3) gives
[quote] A x1 e V: E seV: Ay: (yes <-> y=x1)
[/quote]
No you need to prove that Phi(y,x1) is not paradoxical for all x1, can
you prove that.
[quote]so the set of all singleton's simply do not
have singleton set that contains it as its sole member. However this
set is of course not well founded since it can contain
members like {{U}} were U is the set of all sets.
If U is the set of all sets, what is the intended interpretation of V?
The rules of pairing, union, power, seem to be restricted to well
founded sets, or at most to sets that permit no circular membership
(set in which all members of their transitive closure are not in their
transitive closure).
Infinity is a theorem here also.
Since we cannot use formulas like ~yey, y is ordinal, y is well
founded, etc..., since power is defined for well founded sets, then it
is obvious that Russell's, Burali-Forti, and Cantor's paradox are all
avoided.
It is also interesting to see that the set of all finite sets in this
theory is a set!
However this theory turns to be very complex when go out of well
founded sets, and it is very difficult to set forth any general rules
in such a harsh grounds.
Zuhair[/quote] |
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| zuhair... |
| Posted: Sun Nov 01, 2009 5:11 am |
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Guest
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On Nov 1, 7:53 am, hagman <goo... at (no spam) von-eitzen.de> wrote:
[quote]On 1 Nov., 02:21, zuhair <zaljo... at (no spam) yahoo.com> wrote:
I call this theory "logical set theory" simply because the main
motivation behind it is "logical" in the sence that this theory
follows a logical approach to avoid paradoxes.
LST is the set of all sentences entailed (from FOL with identity,
membership and the primitive constant V) by the following non logical
axioms.
1) Extensionality: For all z ( z e x <-> z e y ) -> x=y
2) Class Comprehension:if Phi is a formula 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.
3) Set Comprehension: IF Phi(y,x1,...,xn) is a formula which do not
use V, and in which y,x1,...,xn are its sole free variables, and in
which s is not free, and if Q1,...,Qm are all the sub-formulas of Phi
in which y or c1 or...or cm are free, having no parameters other than
those in Phi, then
For all c1,...,cm& x1 e V,...,xn e V (
~(Q1(c1) & c1={y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm(cm) & cm={y|Qm}& For all y ( Qm -> ~yey ))
-> Exist a set s for all y ( y e s <-> Phi(y,x1,...,xn ) ).
I assume "s is a set" is the same as "s e V" throughout ?
are axioms.
4) Anti-foundation: Exist x: x e x
5) Transitive: For all x , y ( y e x & x e V -> y e V ).
Theory definition finished/
Zuhair
It is interesting to see that even Leśniewski's paradox is not a
paradox in this theory:
Leśniewski's paradox is about the set of all singletons that are not
members of their sole members, now a singleton set containing this set
would be clearly paradoxical.
The set of all singleton's do exist in this theory definitely,
Hm, how do you see this?
Probably by 3) with
Phi(x):<=> E y: A z: ( yex & ( zex -> z=y ))
I'm not completely sure what is meant with "all the subformulas of
Phi"
Maybe
Q1(x,y):<=> A z: ( yex & ( zex -> z=y ))
Q2(x,y,z):<=> ( yex & ( zex -> z=y ))
and so on or maybe just the "first level" Q1.
Anyway, it appears that the Qi should only have one free variable(?)
so there seems to be no applicable subformula, i.e. m=n=0, hence
the corresponsing instance of 3) is actually
Exist a set s for all y ( y e s <-> Phi(y) ).
Is this the correct reasoning or could you elaborate more on
the subformulas?
but yet
it doesn't lead to any paradox, since the singleton axiom is not a
theorem in this theory,
Do you mean that
A xeV: E seV: A y: (yes <-> y=x)
is not a theorem?
I'd try with Phi(y,x1):<-> y=x1
In my eyes, Phi has no subformulas at all (m=0, n=1), thus 3) gives
A x1 e V: E seV: Ay: (yes <-> y=x1)
so the set of all singleton's simply do not
have singleton set that contains it as its sole member. However this
set is of course not well founded since it can contain
members like {{U}} were U is the set of all sets.
If U is the set of all sets, what is the intended interpretation of V?
The rules of pairing, union, power, seem to be restricted to well
founded sets, or at most to sets that permit no circular membership
(set in which all members of their transitive closure are not in their
transitive closure).
Infinity is a theorem here also.
Since we cannot use formulas like ~yey, y is ordinal, y is well
founded, etc..., since power is defined for well founded sets, then it
is obvious that Russell's, Burali-Forti, and Cantor's paradox are all
avoided.
It is also interesting to see that the set of all finite sets in this
theory is a set!
However this theory turns to be very complex when go out of well
founded sets, and it is very difficult to set forth any general rules
in such a harsh grounds.
Zuhair
[/quote]
On second look, you seem to be right! this theory do prove the
existence of a singleton for every set other than Quine's atom.
hmmmm....
So the paradox of singleton do work here. |
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| zuhair... |
| Posted: Sun Nov 01, 2009 5:51 am |
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Guest
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On Nov 1, 10:11 am, zuhair <zaljo... at (no spam) yahoo.com> wrote:
[quote]On Nov 1, 7:53 am, hagman <goo... at (no spam) von-eitzen.de> wrote:
On 1 Nov., 02:21, zuhair <zaljo... at (no spam) yahoo.com> wrote:
I call this theory "logical set theory" simply because the main
motivation behind it is "logical" in the sence that this theory
follows a logical approach to avoid paradoxes.
LST is the set of all sentences entailed (from FOL with identity,
membership and the primitive constant V) by the following non logical
axioms.
1) Extensionality: For all z ( z e x <-> z e y ) -> x=y
2) Class Comprehension:if Phi is a formula 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.
3) Set Comprehension: IF Phi(y,x1,...,xn) is a formula which do not
use V, and in which y,x1,...,xn are its sole free variables, and in
which s is not free, and if Q1,...,Qm are all the sub-formulas of Phi
in which y or c1 or...or cm are free, having no parameters other than
those in Phi, then
For all c1,...,cm& x1 e V,...,xn e V (
~(Q1(c1) & c1={y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm(cm) & cm={y|Qm}& For all y ( Qm -> ~yey ))
-> Exist a set s for all y ( y e s <-> Phi(y,x1,...,xn ) ).
I assume "s is a set" is the same as "s e V" throughout ?
are axioms.
4) Anti-foundation: Exist x: x e x
5) Transitive: For all x , y ( y e x & x e V -> y e V ).
Theory definition finished/
Zuhair
It is interesting to see that even Leśniewski's paradox is not a
paradox in this theory:
Leśniewski's paradox is about the set of all singletons that are not
members of their sole members, now a singleton set containing this set
would be clearly paradoxical.
The set of all singleton's do exist in this theory definitely,
Hm, how do you see this?
Probably by 3) with
Phi(x):<=> E y: A z: ( yex & ( zex -> z=y ))
I'm not completely sure what is meant with "all the subformulas of
Phi"
Maybe
Q1(x,y):<=> A z: ( yex & ( zex -> z=y ))
Q2(x,y,z):<=> ( yex & ( zex -> z=y ))
and so on or maybe just the "first level" Q1.
Anyway, it appears that the Qi should only have one free variable(?)
so there seems to be no applicable subformula, i.e. m=n=0, hence
the corresponsing instance of 3) is actually
Exist a set s for all y ( y e s <-> Phi(y) ).
Is this the correct reasoning or could you elaborate more on
the subformulas?
but yet
it doesn't lead to any paradox, since the singleton axiom is not a
theorem in this theory,
Do you mean that
A xeV: E seV: A y: (yes <-> y=x)
is not a theorem?
I'd try with Phi(y,x1):<-> y=x1
In my eyes, Phi has no subformulas at all (m=0, n=1), thus 3) gives
A x1 e V: E seV: Ay: (yes <-> y=x1)
so the set of all singleton's simply do not
have singleton set that contains it as its sole member. However this
set is of course not well founded since it can contain
members like {{U}} were U is the set of all sets.
If U is the set of all sets, what is the intended interpretation of V?
The rules of pairing, union, power, seem to be restricted to well
founded sets, or at most to sets that permit no circular membership
(set in which all members of their transitive closure are not in their
transitive closure).
Infinity is a theorem here also.
Since we cannot use formulas like ~yey, y is ordinal, y is well
founded, etc..., since power is defined for well founded sets, then it
is obvious that Russell's, Burali-Forti, and Cantor's paradox are all
avoided.
It is also interesting to see that the set of all finite sets in this
theory is a set!
However this theory turns to be very complex when go out of well
founded sets, and it is very difficult to set forth any general rules
in such a harsh grounds.
Zuhair
On second look, you seem to be right! this theory do prove the
existence of a singleton for every set other than Quine's atom.
hmmmm....
So the paradox of singleton do work here.
[/quote]
On third look, it seem that this theory is much more difficult than I
thought.
Let us examine it with some depth.
All what this theory is saying is that if a formula phi define a
certain class x (i.e. we have Exist x for all y ( y e x iff y e V and
phi(y))
then if phi hold for x also, and at the same time we have
for all y(phi(y) -> ~yey), then phi is said to be paradoxical
and it should not be used weather as a formula or a subformula.
So any formula that is not paradoxical in the sense above, then it can
define a set x , i.e we can have x e V and
for all y ( y e x <-> Phi(y) ).
That is essentially the basic idea behind this theory.
Now lets examine the paradox of Singletons.
so the formula would be
Phi(x) <-> x is singleton and for all y ( y e x -> ~ x e y )
Now lets take the subformula
Q(x) <-> for all y ( y e x -> ~ x e y )
so Q is simply the formula of a class being not a member of any of its
member.
Now lets see if Q(x) is paradoxical or not?
first we do have for all x ( Q(x) -> ~ x e x ) that is clear!
Now lets examine the second characteristic
{x|Q(x)} which is the set of all sets that are not members of their
members. now is this set a member of a member of it or not?
Now it is obvious that {x|Q(x)} is not a member of V, otherwise it
would be paradoxical class, so by then we have this class as not
member of its members but at the same time it is not in V.
This means that the subformula for all y ( y e x -> ~ x e y )
cannot be used since it is paradoxical.
On the other hand we might use another formula for the paradox of
singletons that is
Phi(x) <-> x is singleton and Exist y ( y e x & ~ x e y )
Now lets take the subformula
Q(x) <-> Exist y ( y e x & ~ x e y )
this will mount to Phi being equivalent to saying that x is singleton
other than Quine's atom, anyhow.
However here we do have ~ for all x ( Q(x) -> ~yey )
which is obvious .
this mean that Q(x) can be used as a subformula.
thus the theory is inconsistent!
Zuhair |
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| master1729... |
| Posted: Sun Nov 01, 2009 11:06 am |
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Guest
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zuhair wrote :
[quote]I call this theory "logical set theory" simply
because the main
motivation behind it is "logical" in the sence that
this theory
follows a logical approach to avoid paradoxes.
[/quote]
(snip)
[quote]4) Anti-foundation: Exist x: x e x
5) Transitive: For all x , y ( y e x & x e V -> y
e V ).
Theory definition finished/
Zuhair
[/quote]
it seems - at first sight - that 4) and 5) are consistant with my own set theory.
[quote]
It is interesting to see that even Leśniewski's
paradox is not a
paradox in this theory:
Leśniewski's paradox is about the set of all
singletons that are not
members of their sole members, now a singleton set
containing this set
would be clearly paradoxical.
[/quote]
lesniewski's paradox does not occur within my set theory either.
[x] = x
[quote]
The set of all singleton's do exist in this theory
definitely, but yet
it doesn't lead to any paradox, since the singleton
axiom is not a
theorem in this theory, so the set of all singleton's
simply do not
have singleton set that contains it as its sole
member. However this
set is of course not well founded since it can
contain
members like {{U}} were U is the set of all sets.
[/quote]
that is no problem when u put x = [x].
[quote]
Since we cannot use formulas like ~yey, y is ordinal,
y is well
founded, etc..., since power is defined for well
founded sets, then it
is obvious that Russell's, Burali-Forti, and Cantor's
paradox are all
avoided.
[/quote]
right !
[quote]
It is also interesting to see that the set of all
finite sets in this
theory is a set!
[/quote]
maybe , maybe not.
[quote]
However this theory turns to be very complex when go
out of well
founded sets, and it is very difficult to set forth
any general rules
in such a harsh grounds.
[/quote]
my ' theory ' is simpler ?
[quote]
Zuhair
[/quote]
finally we meet x)
lwalke3 talks often about you.
im the other guy he often mentions , perhaps you heard / read about me too.
nice to "meet you"
regards
tommy1729 |
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| master1729... |
| Posted: Sun Nov 01, 2009 11:10 am |
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Guest
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zuhair wrote :
[quote]On Nov 1, 10:11 am, zuhair <zaljo... at (no spam) yahoo.com
wrote:
On Nov 1, 7:53 am, hagman <goo... at (no spam) von-eitzen.de
wrote:
On 1 Nov., 02:21, zuhair <zaljo... at (no spam) yahoo.com
wrote:
I call this theory "logical set theory" simply
because the main
motivation behind it is "logical" in the sence
that this theory
follows a logical approach to avoid paradoxes.
LST is the set of all sentences entailed (from
FOL with identity,
membership and the primitive constant V) by the
following non logical
axioms.
1) Extensionality: For all z ( z e x <-> z e y
) -> x=y
2) Class Comprehension:if Phi is a formula 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.
3) Set Comprehension: IF Phi(y,x1,...,xn) is a
formula which do not
use V, and in which y,x1,...,xn are its sole
free variables, and in
which s is not free, and if Q1,...,Qm are all
the sub-formulas of Phi
in which y or c1 or...or cm are free, having no
parameters other than
those in Phi, then
For all c1,...,cm& x1 e V,...,xn e V (
~(Q1(c1) & c1={y|Q1}& For all y ( Q1 -> ~yey
)),...,
~(Qm(cm) & cm={y|Qm}& For all y ( Qm -> ~yey ))
-> Exist a set s for all y ( y e s <-
Phi(y,x1,...,xn ) ).
I assume "s is a set" is the same as "s e V"
throughout ?
are axioms.
4) Anti-foundation: Exist x: x e x
5) Transitive: For all x , y ( y e x & x e V
-> y e V ).
Theory definition finished/
Zuhair
It is interesting to see that even Leśniewski's
paradox is not a
paradox in this theory:
Leśniewski's paradox is about the set of all
singletons that are not
members of their sole members, now a singleton
set containing this set
would be clearly paradoxical.
The set of all singleton's do exist in this
theory definitely,
Hm, how do you see this?
Probably by 3) with
Phi(x):<=> E y: A z: ( yex & ( zex -> z=y ))
I'm not completely sure what is meant with "all
the subformulas of
Phi"
Maybe
Q1(x,y):<=> A z: ( yex & ( zex -> z=y ))
Q2(x,y,z):<=> ( yex & ( zex -> z=y ))
and so on or maybe just the "first level" Q1.
Anyway, it appears that the Qi should only have
one free variable(?)
so there seems to be no applicable subformula,
i.e. m=n=0, hence
the corresponsing instance of 3) is actually
Exist a set s for all y ( y e s <-> Phi(y) ).
Is this the correct reasoning or could you
elaborate more on
the subformulas?
but yet
it doesn't lead to any paradox, since the
singleton axiom is not a
theorem in this theory,
Do you mean that
A xeV: E seV: A y: (yes <-> y=x)
is not a theorem?
I'd try with Phi(y,x1):<-> y=x1
In my eyes, Phi has no subformulas at all (m=0,
n=1), thus 3) gives
A x1 e V: E seV: Ay: (yes <-> y=x1)
so the set of all singleton's simply do not
have singleton set that contains it as its sole
member. However this
set is of course not well founded since it can
contain
members like {{U}} were U is the set of all
sets.
If U is the set of all sets, what is the intended
interpretation of V?
The rules of pairing, union, power, seem to be
restricted to well
founded sets, or at most to sets that permit no
circular membership
(set in which all members of their transitive
closure are not in their
transitive closure).
Infinity is a theorem here also.
Since we cannot use formulas like ~yey, y is
ordinal, y is well
founded, etc..., since power is defined for
well founded sets, then it
is obvious that Russell's, Burali-Forti, and
Cantor's paradox are all
avoided.
It is also interesting to see that the set of
all finite sets in this
theory is a set!
However this theory turns to be very complex
when go out of well
founded sets, and it is very difficult to set
forth any general rules
in such a harsh grounds.
Zuhair
On second look, you seem to be right! this theory
do prove the
existence of a singleton for every set other than
Quine's atom.
hmmmm....
So the paradox of singleton do work here.
On third look, it seem that this theory is much more
difficult than I
thought.
Let us examine it with some depth.
All what this theory is saying is that if a formula
phi define a
certain class x (i.e. we have Exist x for all y ( y e
x iff y e V and
phi(y))
then if phi hold for x also, and at the same time we
have
for all y(phi(y) -> ~yey), then phi is said to be
paradoxical
and it should not be used weather as a formula or a
subformula.
So any formula that is not paradoxical in the sense
above, then it can
define a set x , i.e we can have x e V and
for all y ( y e x <-> Phi(y) ).
That is essentially the basic idea behind this
theory.
Now lets examine the paradox of Singletons.
so the formula would be
Phi(x) <-> x is singleton and for all y ( y e x -> ~
x e y )
Now lets take the subformula
Q(x) <-> for all y ( y e x -> ~ x e y )
so Q is simply the formula of a class being not a
member of any of its
member.
Now lets see if Q(x) is paradoxical or not?
first we do have for all x ( Q(x) -> ~ x e x ) that
is clear!
Now lets examine the second characteristic
{x|Q(x)} which is the set of all sets that are not
members of their
members. now is this set a member of a member of it
or not?
Now it is obvious that {x|Q(x)} is not a member of V,
otherwise it
would be paradoxical class, so by then we have this
class as not
member of its members but at the same time it is not
in V.
This means that the subformula for all y ( y e x -
~ x e y )
cannot be used since it is paradoxical.
On the other hand we might use another formula for
the paradox of
singletons that is
Phi(x) <-> x is singleton and Exist y ( y e x & ~ x e
y )
Now lets take the subformula
Q(x) <-> Exist y ( y e x & ~ x e y )
this will mount to Phi being equivalent to saying
that x is singleton
other than Quine's atom, anyhow.
However here we do have ~ for all x ( Q(x) -> ~yey )
which is obvious .
this mean that Q(x) can be used as a subformula.
thus the theory is inconsistent!
Zuhair
[/quote]
may i recommend some of my own ideas :
[x] = x
~[] = []
3-valued logic
regards
tommy1729 |
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| zuhair... |
| Posted: Sun Nov 01, 2009 12:03 pm |
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Guest
|
On Nov 1, 4:10 pm, master1729 <tommy1... at (no spam) gmail.com> wrote:
[quote]zuhair wrote :
On Nov 1, 10:11 am, zuhair <zaljo... at (no spam) yahoo.com
wrote:
On Nov 1, 7:53 am, hagman <goo... at (no spam) von-eitzen.de
wrote:
On 1 Nov., 02:21, zuhair <zaljo... at (no spam) yahoo.com
wrote:
I call this theory "logical set theory" simply
because the main
motivation behind it is "logical" in the sence
that this theory
follows a logical approach to avoid paradoxes.
LST is the set of all sentences entailed (from
FOL with identity,
membership and the primitive constant V) by the
following non logical
axioms.
1) Extensionality: For all z ( z e x <-> z e y
) -> x=y
2) Class Comprehension:if Phi is a formula 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.
3) Set Comprehension: IF Phi(y,x1,...,xn) is a
formula which do not
use V, and in which y,x1,...,xn are its sole
free variables, and in
which s is not free, and if Q1,...,Qm are all
the sub-formulas of Phi
in which y or c1 or...or cm are free, having no
parameters other than
those in Phi, then
For all c1,...,cm& x1 e V,...,xn e V (
~(Q1(c1) & c1={y|Q1}& For all y ( Q1 -> ~yey
)),...,
~(Qm(cm) & cm={y|Qm}& For all y ( Qm -> ~yey ))
-> Exist a set s for all y ( y e s <-
Phi(y,x1,...,xn ) ).
I assume "s is a set" is the same as "s e V"
throughout ?
are axioms.
4) Anti-foundation: Exist x: x e x
5) Transitive: For all x , y ( y e x & x e V
-> y e V ).
Theory definition finished/
Zuhair
It is interesting to see that even Leśniewski's
paradox is not a
paradox in this theory:
Leśniewski's paradox is about the set of all
singletons that are not
members of their sole members, now a singleton
set containing this set
would be clearly paradoxical.
The set of all singleton's do exist in this
theory definitely,
Hm, how do you see this?
Probably by 3) with
Phi(x):<=> E y: A z: ( yex & ( zex -> z=y ))
I'm not completely sure what is meant with "all
the subformulas of
Phi"
Maybe
Q1(x,y):<=> A z: ( yex & ( zex -> z=y ))
Q2(x,y,z):<=> ( yex & ( zex -> z=y ))
and so on or maybe just the "first level" Q1.
Anyway, it appears that the Qi should only have
one free variable(?)
so there seems to be no applicable subformula,
i.e. m=n=0, hence
the corresponsing instance of 3) is actually
Exist a set s for all y ( y e s <-> Phi(y) ).
Is this the correct reasoning or could you
elaborate more on
the subformulas?
but yet
it doesn't lead to any paradox, since the
singleton axiom is not a
theorem in this theory,
Do you mean that
A xeV: E seV: A y: (yes <-> y=x)
is not a theorem?
I'd try with Phi(y,x1):<-> y=x1
In my eyes, Phi has no subformulas at all (m=0,
n=1), thus 3) gives
A x1 e V: E seV: Ay: (yes <-> y=x1)
so the set of all singleton's simply do not
have singleton set that contains it as its sole
member. However this
set is of course not well founded since it can
contain
members like {{U}} were U is the set of all
sets.
If U is the set of all sets, what is the intended
interpretation of V?
The rules of pairing, union, power, seem to be
restricted to well
founded sets, or at most to sets that permit no
circular membership
(set in which all members of their transitive
closure are not in their
transitive closure).
Infinity is a theorem here also.
Since we cannot use formulas like ~yey, y is
ordinal, y is well
founded, etc..., since power is defined for
well founded sets, then it
is obvious that Russell's, Burali-Forti, and
Cantor's paradox are all
avoided.
It is also interesting to see that the set of
all finite sets in this
theory is a set!
However this theory turns to be very complex
when go out of well
founded sets, and it is very difficult to set
forth any general rules
in such a harsh grounds.
Zuhair
On second look, you seem to be right! this theory
do prove the
existence of a singleton for every set other than
Quine's atom.
hmmmm....
So the paradox of singleton do work here.
On third look, it seem that this theory is much more
difficult than I
thought.
Let us examine it with some depth.
All what this theory is saying is that if a formula
phi define a
certain class x (i.e. we have Exist x for all y ( y e
x iff y e V and
phi(y))
then if phi hold for x also, and at the same time we
have
for all y(phi(y) -> ~yey), then phi is said to be
paradoxical
and it should not be used weather as a formula or a
subformula.
So any formula that is not paradoxical in the sense
above, then it can
define a set x , i.e we can have x e V and
for all y ( y e x <-> Phi(y) ).
That is essentially the basic idea behind this
theory.
Now lets examine the paradox of Singletons.
so the formula would be
Phi(x) <-> x is singleton and for all y ( y e x -> ~
x e y )
Now lets take the subformula
Q(x) <-> for all y ( y e x -> ~ x e y )
so Q is simply the formula of a class being not a
member of any of its
member.
Now lets see if Q(x) is paradoxical or not?
first we do have for all x ( Q(x) -> ~ x e x ) that
is clear!
Now lets examine the second characteristic
{x|Q(x)} which is the set of all sets that are not
members of their
members. now is this set a member of a member of it
or not?
Now it is obvious that {x|Q(x)} is not a member of V,
otherwise it
would be paradoxical class, so by then we have this
class as not
member of its members but at the same time it is not
in V.
This means that the subformula for all y ( y e x -
~ x e y )
cannot be used since it is paradoxical.
On the other hand we might use another formula for
the paradox of
singletons that is
Phi(x) <-> x is singleton and Exist y ( y e x & ~ x e
y )
Now lets take the subformula
Q(x) <-> Exist y ( y e x & ~ x e y )
this will mount to Phi being equivalent to saying
that x is singleton
other than Quine's atom, anyhow.
However here we do have ~ for all x ( Q(x) -> ~yey )
which is obvious .
this mean that Q(x) can be used as a subformula.
thus the theory is inconsistent!
Zuhair
may i recommend some of my own ideas :
[x] = x
~[] = []
3-valued logic
regards
tommy1729
[/quote]
The idea of [x]=x is Mereological, or if you read my Flat set theory
that I wrote perhaps year a go, you'll see such a thing
But I don't know about ~[] = []
this is a little difficult for me.
nice to meet you.
Zuhair |
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| zuhair... |
| Posted: Sun Nov 01, 2009 1:47 pm |
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Guest
|
On Nov 1, 4:06 pm, master1729 <tommy1... at (no spam) gmail.com> wrote:
[quote]zuhair wrote :
I call this theory "logical set theory" simply
because the main
motivation behind it is "logical" in the sence that
this theory
follows a logical approach to avoid paradoxes.
(snip)
4) Anti-foundation: Exist x: x e x
5) Transitive: For all x , y ( y e x & x e V -> y
e V ).
Theory definition finished/
Zuhair
it seems - at first sight - that 4) and 5) are consistant with my own set theory.
It is interesting to see that even Leśniewski's
paradox is not a
paradox in this theory:
Leśniewski's paradox is about the set of all
singletons that are not
members of their sole members, now a singleton set
containing this set
would be clearly paradoxical.
lesniewski's paradox does not occur within my set theory either.
[x] = x
The set of all singleton's do exist in this theory
definitely, but yet
it doesn't lead to any paradox, since the singleton
axiom is not a
theorem in this theory, so the set of all singleton's
simply do not
have singleton set that contains it as its sole
member. However this
set is of course not well founded since it can
contain
members like {{U}} were U is the set of all sets.
that is no problem when u put x = [x].
Since we cannot use formulas like ~yey, y is ordinal,
y is well
founded, etc..., since power is defined for well
founded sets, then it
is obvious that Russell's, Burali-Forti, and Cantor's
paradox are all
avoided.
right !
It is also interesting to see that the set of all
finite sets in this
theory is a set!
maybe , maybe not.
However this theory turns to be very complex when go
out of well
founded sets, and it is very difficult to set forth
any general rules
in such a harsh grounds.
my ' theory ' is simpler ?
Zuhair
finally we meet x)
lwalke3 talks often about you.
im the other guy he often mentions , perhaps you heard / read about me too.
nice to "meet you"
regards
tommy1729
[/quote]
Yea, I heard of you of course, the master yea.
Really nice to meet you.
Can you provide me with a link to your theory.
Zuhair |
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