| |
 |
|
| Science Forum Index » Logic Forum » Towards avoiding paradoxes with set theory.... |
|
Page 1 of 4 Goto page 1, 2, 3, 4 Next |
|
| Author |
Message |
| zuhair... |
 Posted: Sat Oct 31, 2009 7:16 am |
|
|
|
Guest
|
Hi all,
The well known paradoxes of set theory:Russell's, Burali-
Forti,Leśniewski's and Cantor's, all can be shown to be reduced to the
following argument.
IF we work in a theory in first order logic with identity and the
primitive constant V , and having the axioms of Extensionality and
class comprehension (as present in Ackermanns' set theory)
let me rewrite axiom schema of 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.
To simplify matters lets use the set builder abstraction { | } defined
below:
Define: x={y|Phi} <-> For all y ( y e x <-> (y e V & Phi) )
So class comprehension define classes that are subclasses of V, some
of these classes would be sets i.e members of V, while others will not
be sets i.e. not members of V, and we do not know if V itself would be
a set or not.
For the known paradoxes of set theory, we are sure of one fact, that
is not all formulas of First order logic with identity and V are
suitable to define 'set's. So we need to choose suitable formulas that
can define sets.
We need to define sets in a semi-Naive manner using the formula below:
Exist a set x for all y ( y e x <-> Phi )
But before that we need to examine the formulas that lead to paradoxes
if we use them in the above sentence ,and see what is the basic
characteristic that made them paradoxical.
I observed the following paradoxical characteristic.
If Phi contain any sub-formula Q that have BOTH of the following
characteristics:
1.Q{y|Q}
2.For all y ( Q -> ~yey )
then Phi is paradoxical, and IF Phi is paradoxical then there is a sub-
formula Q of Phi such that 1. and 2. above.
Lets see that closely with Russell's paradox:
take Q to be ~xex
we have ~ {y|~yey} e {y|~yey} i.e we have Q{y|Q}
and we have: for all x ( ~yey -> ~yey )
so ~xex is obviously paradoxical.
Take another example: Let Q to be " x is ordinal"
we have {x | x is Ordinal} is Ordinal
and we have: for all x ( x is ordinal -> ~ x e x )
so the formula " x is ordinal" is paradoxical.
Also take the formula " x is well founded"
we have: {x| x is well founded} is well founded
and we have: for all x ( x is well founded -> ~ x e x ).
Same to be said about the other paradoxes.
Now what if we simply axiomatize a version of Naive comprehension that
forbid the use of paradoxical formulas outlined above.
Axiom schema of Set comprehension: If Phi is not a paradoxical
formula, and in which x is not free, then all closures of
Exist a set x for all y ( y e x <-> Phi )
are axioms.
the above schema simply states that any x={y|Phi} is a set if Phi is
non paradoxical , in which x is not free.
Now for all sets in which every non empty subset of them is disjoint
of them, then such sets would be well founded sets, now these sets
would have all ZF set theory axioms.
Lets add a fourth axiom scheme which is
Axiom: Exist x : x e x
Now this set theory with only these four axiom schemes would be one
which has the set of all sets in it, i.e we do have
V e V here.
take phi to be y=y
then we have ~ for all y ( y=y -> ~yey )
thus y=y is not a paradoxical formula
then it can be used in set comprehension.
we can also use the formula y e y in set comprehension, since it is
also not paradoxical, etc...
The interesting thing here is that if we define a set to be a member
of V that is well founded, then all ZF axioms would follow.
However we do need the bigger sets like V and the others, so it is
better to keep the definition of set as a member of V.
I think if this set theory is not inconsistent, then it would be
maximal in terms of construction, since we can construct all ZF sets
and in addition to it we'll have universal sets also.
So let me Define theory T.
T 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 sub-formulas of Phi in
which y is free, and their parameters are subset of the parameters of
Phi, then
For all x1 e V,...,xn e V (
~(Q1{y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm{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
Theory definition finished/
Zuhair |
|
|
| Back to top |
|
|
|
| Charlie-Boo... |
| Posted: Sat Oct 31, 2009 8:41 am |
|
|
|
Guest
|
On Oct 31, 1:16Â pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
[quote]Hi all,
The well known paradoxes of set theory:Russell's, Burali-
Forti,Leśniewski's and Cantor's, all can be shown to be reduced to the
following argument.
[/quote]
You are trying to find a value N such that {x|P(N,x) holds} equals the
set {x|~P(x,x) holds} for some P.
Any other questions?
C-B
[quote]IF we work in a theory in first order logic with identity and the
primitive constant V , and having the axioms of Extensionality and
class comprehension (as present in Ackermanns' set theory)
let me rewrite axiom schema of 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.
To simplify matters lets use the set builder abstraction { | } defined
below:
Define: x={y|Phi} Â <-> For all y ( y e x <-> (y e V & Phi) )
So class comprehension define classes that are subclasses of V, some
of these classes would be sets i.e members of V, while others will not
be sets i.e. not members of V, and we do not know if V itself would be
a set or not.
For the known paradoxes of set theory, we are sure of one fact, that
is not all formulas of First order logic with identity and V are
suitable to define 'set's. So we need to choose suitable formulas that
can define sets.
We need to define sets in a semi-Naive manner using the formula below:
Exist a set x for all y ( y e x <-> Phi )
But before that we need to examine the formulas that lead to paradoxes
if we use them in the above sentence ,and see what is the basic
characteristic that made them paradoxical.
I observed the following paradoxical characteristic.
If Phi contain any sub-formula Q that have BOTH of the following
characteristics:
1.Q{y|Q}
2.For all y ( Q -> ~yey )
then Phi is paradoxical, and IF Phi is paradoxical then there is a sub-
formula Q of Phi such that 1. and 2. above.
Lets see that closely with Russell's paradox:
 take Q to be ~xex
we have ~ {y|~yey} e {y|~yey} Â i.e we have Q{y|Q}
and we have: for all x ( ~yey -> ~yey )
so ~xex is obviously paradoxical.
Take another example: Let Q to be " x is ordinal"
we have  {x | x is Ordinal} is Ordinal
and we have: for all x ( x is ordinal -> ~ x e x )
so the formula " x is ordinal" is paradoxical.
Also take the formula " x is well founded"
we have: {x| x is well founded} is well founded
and we have: for all x ( x is well founded -> ~ x e x ).
Same to be said about the other paradoxes.
Now what if we simply axiomatize a version of Naive comprehension that
forbid the use of paradoxical formulas outlined above.
Axiom schema of Set comprehension: If Phi is not a paradoxical
formula, and in which x is not free, then all closures of
Exist a set x for all y ( y e x <-> Phi )
are axioms.
the above schema simply states that any x={y|Phi} is a set if Phi is
non paradoxical , in which x is not free.
Now for all sets in which every non empty subset of them is disjoint
of them, then such sets would be well founded sets, now these sets
would have all ZF set theory axioms.
Lets add a fourth axiom scheme which is
Axiom: Exist x : x e x
Now this set theory with only these four axiom schemes would be one
which has the set of all sets in it, i.e we do have
V e V here.
take phi to be y=y
then we have ~ for all y ( y=y -> ~yey )
thus y=y is not a paradoxical formula
then it can be used in set comprehension.
we can also use the formula y e y in set comprehension, since it is
also not paradoxical, etc...
The interesting thing here is that if we define a set to be a member
of V that is well founded, then all ZF axioms would follow.
However we do need the bigger sets like V and the others, so it is
better to keep the definition of set as a member of V.
I think if this set theory is not inconsistent, then it would be
maximal in terms of construction, since we can construct all ZF sets
and in addition to it we'll have universal sets also.
So let me Define theory T.
T 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 ) Â -> 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 sub-formulas of Phi in
which y is free, and their parameters are subset of the parameters of
Phi, then
For all x1 e V,...,xn e V (
~(Q1{y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm{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
Theory definition finished/
Zuhair[/quote] |
|
|
| Back to top |
|
|
|
| zuhair... |
| Posted: Sat Oct 31, 2009 9:01 am |
|
|
|
Guest
|
On Oct 31, 12:16Â pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
[quote]Hi all,
The well known paradoxes of set theory:Russell's, Burali-
Forti,Leśniewski's and Cantor's, all can be shown to be reduced to the
following argument.
IF we work in a theory in first order logic with identity and the
primitive constant V , and having the axioms of Extensionality and
class comprehension (as present in Ackermanns' set theory)
let me rewrite axiom schema of 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.
To simplify matters lets use the set builder abstraction { | } defined
below:
Define: x={y|Phi} Â <-> For all y ( y e x <-> (y e V & Phi) )
So class comprehension define classes that are subclasses of V, some
of these classes would be sets i.e members of V, while others will not
be sets i.e. not members of V, and we do not know if V itself would be
a set or not.
For the known paradoxes of set theory, we are sure of one fact, that
is not all formulas of First order logic with identity and V are
suitable to define 'set's. So we need to choose suitable formulas that
can define sets.
We need to define sets in a semi-Naive manner using the formula below:
Exist a set x for all y ( y e x <-> Phi )
But before that we need to examine the formulas that lead to paradoxes
if we use them in the above sentence ,and see what is the basic
characteristic that made them paradoxical.
I observed the following paradoxical characteristic.
If Phi contain any sub-formula Q that have BOTH of the following
characteristics:
1.Q{y|Q}
2.For all y ( Q -> ~yey )
then Phi is paradoxical, and IF Phi is paradoxical then there is a sub-
formula Q of Phi such that 1. and 2. above.
Lets see that closely with Russell's paradox:
 take Q to be ~xex
we have ~ {y|~yey} e {y|~yey} Â i.e we have Q{y|Q}
and we have: for all x ( ~yey -> ~yey )
so ~xex is obviously paradoxical.
Take another example: Let Q to be " x is ordinal"
we have  {x | x is Ordinal} is Ordinal
and we have: for all x ( x is ordinal -> ~ x e x )
so the formula " x is ordinal" is paradoxical.
Also take the formula " x is well founded"
we have: {x| x is well founded} is well founded
and we have: for all x ( x is well founded -> ~ x e x ).
Same to be said about the other paradoxes.
Now what if we simply axiomatize a version of Naive comprehension that
forbid the use of paradoxical formulas outlined above.
Axiom schema of Set comprehension: If Phi is not a paradoxical
formula, and in which x is not free, then all closures of
Exist a set x for all y ( y e x <-> Phi )
are axioms.
the above schema simply states that any x={y|Phi} is a set if Phi is
non paradoxical , in which x is not free.
Now for all sets in which every non empty subset of them is disjoint
of them, then such sets would be well founded sets, now these sets
would have all ZF set theory axioms.
Lets add a fourth axiom scheme which is
Axiom: Exist x : x e x
Now this set theory with only these four axiom schemes would be one
which has the set of all sets in it, i.e we do have
V e V here.
take phi to be y=y
then we have ~ for all y ( y=y -> ~yey )
thus y=y is not a paradoxical formula
then it can be used in set comprehension.
we can also use the formula y e y in set comprehension, since it is
also not paradoxical, etc...
The interesting thing here is that if we define a set to be a member
of V that is well founded, then all ZF axioms would follow.
However we do need the bigger sets like V and the others, so it is
better to keep the definition of set as a member of V.
I think if this set theory is not inconsistent, then it would be
maximal in terms of construction, since we can construct all ZF sets
and in addition to it we'll have universal sets also.
So let me Define theory T.
T 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 ) Â -> 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 sub-formulas of Phi in
which y is free, and their parameters are subset of the parameters of
Phi, then
For all x1 e V,...,xn e V (
~(Q1{y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm{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
[/quote]
No there is axiom 5)
Axiom 5) V is transitive.
i.e. any member of any member of v is a member of V.
[quote]
Theory definition finished/
Zuhair[/quote] |
|
|
| Back to top |
|
|
|
| zuhair... |
| Posted: Sat Oct 31, 2009 9:02 am |
|
|
|
Guest
|
On Oct 31, 1:41Â pm, Charlie-Boo <shymath... at (no spam) gmail.com> wrote:
[quote]On Oct 31, 1:16Â pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
Hi all,
The well known paradoxes of set theory:Russell's, Burali-
Forti,Leśniewski's and Cantor's, all can be shown to be reduced to the
following argument.
You are trying to find a value N such that {x|P(N,x) holds} equals the
set {x|~P(x,x) holds} for some P.
[/quote]
No.
[quote]Any other questions?
C-B
IF we work in a theory in first order logic with identity and the
primitive constant V , and having the axioms of Extensionality and
class comprehension (as present in Ackermanns' set theory)
let me rewrite axiom schema of 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.
To simplify matters lets use the set builder abstraction { | } defined
below:
Define: x={y|Phi} Â <-> For all y ( y e x <-> (y e V & Phi) )
So class comprehension define classes that are subclasses of V, some
of these classes would be sets i.e members of V, while others will not
be sets i.e. not members of V, and we do not know if V itself would be
a set or not.
For the known paradoxes of set theory, we are sure of one fact, that
is not all formulas of First order logic with identity and V are
suitable to define 'set's. So we need to choose suitable formulas that
can define sets.
We need to define sets in a semi-Naive manner using the formula below:
Exist a set x for all y ( y e x <-> Phi )
But before that we need to examine the formulas that lead to paradoxes
if we use them in the above sentence ,and see what is the basic
characteristic that made them paradoxical.
I observed the following paradoxical characteristic.
If Phi contain any sub-formula Q that have BOTH of the following
characteristics:
1.Q{y|Q}
2.For all y ( Q -> ~yey )
then Phi is paradoxical, and IF Phi is paradoxical then there is a sub-
formula Q of Phi such that 1. and 2. above.
Lets see that closely with Russell's paradox:
 take Q to be ~xex
we have ~ {y|~yey} e {y|~yey} Â i.e we have Q{y|Q}
and we have: for all x ( ~yey -> ~yey )
so ~xex is obviously paradoxical.
Take another example: Let Q to be " x is ordinal"
we have  {x | x is Ordinal} is Ordinal
and we have: for all x ( x is ordinal -> ~ x e x )
so the formula " x is ordinal" is paradoxical.
Also take the formula " x is well founded"
we have: {x| x is well founded} is well founded
and we have: for all x ( x is well founded -> ~ x e x ).
Same to be said about the other paradoxes.
Now what if we simply axiomatize a version of Naive comprehension that
forbid the use of paradoxical formulas outlined above.
Axiom schema of Set comprehension: If Phi is not a paradoxical
formula, and in which x is not free, then all closures of
Exist a set x for all y ( y e x <-> Phi )
are axioms.
the above schema simply states that any x={y|Phi} is a set if Phi is
non paradoxical , in which x is not free.
Now for all sets in which every non empty subset of them is disjoint
of them, then such sets would be well founded sets, now these sets
would have all ZF set theory axioms.
Lets add a fourth axiom scheme which is
Axiom: Exist x : x e x
Now this set theory with only these four axiom schemes would be one
which has the set of all sets in it, i.e we do have
V e V here.
take phi to be y=y
then we have ~ for all y ( y=y -> ~yey )
thus y=y is not a paradoxical formula
then it can be used in set comprehension.
we can also use the formula y e y in set comprehension, since it is
also not paradoxical, etc...
The interesting thing here is that if we define a set to be a member
of V that is well founded, then all ZF axioms would follow.
However we do need the bigger sets like V and the others, so it is
better to keep the definition of set as a member of V.
I think if this set theory is not inconsistent, then it would be
maximal in terms of construction, since we can construct all ZF sets
and in addition to it we'll have universal sets also.
So let me Define theory T.
T 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 ) Â -> 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 sub-formulas of Phi in
which y is free, and their parameters are subset of the parameters of
Phi, then
For all x1 e V,...,xn e V (
~(Q1{y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm{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
Theory definition finished/
Zuhair[/quote] |
|
|
| Back to top |
|
|
|
| zuhair... |
| Posted: Sat Oct 31, 2009 9:33 am |
|
|
|
Guest
|
Let me right the theory completely with its five axiom schemes:
T 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 sub-formulas of Phi in
which y is free, and their parameters are subset of the parameters of
Phi, then
For all x1 e V,...,xn e V (
~(Q1{y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm{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 |
|
|
| Back to top |
|
|
|
| zuhair... |
| Posted: Sat Oct 31, 2009 10:10 am |
|
|
|
Guest
|
On Oct 31, 2:33 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
[quote]Let me right the theory completely with its five axiom schemes:
T 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 sub-formulas of Phi in
which y is free, and their parameters are subset of the parameters of
Phi, then
For all x1 e V,...,xn e V (
~(Q1{y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm{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
[/quote]
Actually there is a lot of restrictions on set comprehension, like
parameters being not in V and the formula not using V, I think with
this theory this is not needed. Actually I do believe that we might
dispense with the primitive constant V altogether, and present a
theory in MK fashion with the restriction of not using paradoxical
formulas.
So we can have a theory in FOL with e and =. and define "set" as in
Morse-Kelley set theory as an object that is a member of another
object, in symbols: x is a set <-> Exist y ( x e y )
and have the axiom of Extensionality and the schema of class
comprehension as in Morse-Kelley set theory. and then add the
anti-foundation axiom of Exist x: x e x., and add the following set
comprehension schema.
3) Set Comprehension: IF Phi is a formula in which at least y is free,
and in which x is not free, and if Q1,...,Qm are all
sub-formulas of Phi in which y is free, with no parameter in them
other than those parameters in phi, then all closures of:
~(Q1{y|Q1}& For all y (Q1 -> ~yey))&...&
~(Qm{y|Qm}& For all y (Qm -> ~yey))
-> Exist a set x for all y (y e x <-> Phi).
are axioms.
I think this Morse-Kelley like theory would be sufficient for the
quest of this theory.
The same thing applies here, if we work with well founded sets then it
seems that Morse-Kelley would be a sub-theory of this theory, if we
work with all sets, then perhaps we can have a good theory dealing
with universal sets,while at the same time having Morse-Kelley and
thus ZF as a sub-theory of it.
Zuhair |
|
|
| Back to top |
|
|
|
| Charlie-Boo... |
| Posted: Sat Oct 31, 2009 10:33 am |
|
|
|
Guest
|
On Oct 31, 3:02Â pm, zuhair <zaljo... at (no spam) yahoo.com> wrote:
[quote]On Oct 31, 1:41Â pm, Charlie-Boo <shymath... at (no spam) gmail.com> wrote:
On Oct 31, 1:16Â pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
Hi all,
The well known paradoxes of set theory:Russell's, Burali-
Forti,Leśniewski's and Cantor's, all can be shown to be reduced to the
following argument.
You are trying to find a value N such that {x|P(N,x) holds} equals the
set {x|~P(x,x) holds} for some P.
No.
[/quote]
Counter-example? (Give any specific example - self contained complete
definition - and I will show that it does. I say this because that is
the only axiom in the negative results of Theory of Computation that
is not in the positive results. That is, the single axiom that says
e.g. in the special case the set of programs that do not halt on
themselves is not r.e., is enough to derive all of these results
(along with known properties of every recursive function or r.e.
set.))
C-B
[quote]Any other questions?
C-B[/quote] |
|
|
| Back to top |
|
|
|
| zuhair... |
| Posted: Sat Oct 31, 2009 11:08 am |
|
|
|
Guest
|
On Oct 31, 3:10 pm, zuhair <zaljo... at (no spam) yahoo.com> wrote:
[quote]On Oct 31, 2:33 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
Let me right the theory completely with its five axiom schemes:
T 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 sub-formulas of Phi in
which y is free, and their parameters are subset of the parameters of
Phi, then
For all x1 e V,...,xn e V (
~(Q1{y|Q1}& For all y ( Q1 -> ~yey )),...,
~(Qm{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
Actually there is a lot of restrictions on set comprehension, like
parameters being not in V and the formula not using V, I think with
this theory this is not needed. Actually I do believe that we might
dispense with the primitive constant V altogether, and present a
theory in MK fashion with the restriction of not using paradoxical
formulas.
So we can have a theory in FOL with e and =. and define "set" as in
Morse-Kelley set theory as an object that is a member of another
object, in symbols: x is a set <-> Exist y ( x e y )
and have the axiom of Extensionality and the schema of class
comprehension as in Morse-Kelley set theory. and then add the
anti-foundation axiom of Exist x: x e x., and add the following set
comprehension schema.
3) Set Comprehension: IF Phi is a formula in which at least y is free,
and in which x is not free, and if Q1,...,Qm are all
sub-formulas of Phi in which y is free, with no parameter in them
other than those parameters in phi, then all closures of:
~(Q1{y|Q1}& For all y (Q1 -> ~yey))&...&
~(Qm{y|Qm}& For all y (Qm -> ~yey))
-> Exist a set x for all y (y e x <-> Phi).
are axioms.
I think this Morse-Kelley like theory would be sufficient for the
quest of this theory.
The same thing applies here, if we work with well founded sets then it
seems that Morse-Kelley would be a sub-theory of this theory, if we
work with all sets, then perhaps we can have a good theory dealing
with universal sets,while at the same time having Morse-Kelley and
thus ZF as a sub-theory of it.
Zuhair
[/quote]
I do think now that this theory is weaker than ZF or MK, since it
forbid us from the use of formulas like x is ordinal, etc... in
separation.
Zuhair |
|
|
| Back to top |
|
|
|
| Charlie-Boo... |
| Posted: Sun Nov 01, 2009 2:02 am |
|
|
|
Guest
|
[quote]Zuhair
I do think now that this theory is weaker than ZF or MK, since it
forbid us from the use of formulas like x is ordinal, etc... in
separation.
[/quote]
No it isn't and doesn't - why do you think that?
It is diagonalization.
It is merely a formalization of diagonalization.
-~P/P We cannot represent the negation of the system within the
system.
That is the only axiom needed for negative results. E.g.
-~SE/SE There is no set of all sets that do not contain themselves.
-~YES/YES The set of programs that don't halt Yes is not r.e.
-~TS/TS We cannot define truth in English using English.
where SE, YES and TS are standard i.e.
SE(a,b) "b is an element of a."
YES(a,b) "Turing Machine a with b as input halts yes."
TS(a,b) "English sentence a with noun phrase b substituted for its
pronouns is true."
This provides the only universal justification of its resolution of
the paradoxes (Russell and Liar above) thus satisfying the standard
criteria for correctness.
C-B
[quote]Zuhair
[/quote]
"Zuhair"? It sounds like one of those African natives. Do you have a
bone sticking through your nose? |
|
|
| Back to top |
|
|
|
| Charlie-Boo... |
| Posted: Sun Nov 01, 2009 2:12 am |
|
|
|
Guest
|
On Oct 31, 12:16 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
You can't use set theory because set theory is one of the theories in
which it holds. You have to have something that is not a set. You
have to resist diagonalizing over the system you use. So in general P/
Q does not have a name and cannot be diagonalized. It is not a
recursive function, r.e. set, a function, a set or a system. P and Q
can be sets or relations etc. but P/Q is not a set or class or
function etc. That is the ultimate resolution of all paradoxes (each
created by substitution as I illustrated earlier.)
C-B
> Zuhair |
|
|
| Back to top |
|
|
|
| Libra/Virgo... |
| Posted: Sun Nov 01, 2009 5:34 am |
|
|
|
Guest
|
On Nov 1, 5:12 am, Charlie-Boo <shymath... at (no spam) gmail.com> wrote:
[quote]On Oct 31, 12:16 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
You can't use set theory because set theory is one of the theories in
which it holds. You have to have something that is not a set. You
have to resist diagonalizing over the system you use. So in general P/
Q does not have a name and cannot be diagonalized. It is not a
recursive function, r.e. set, a function, a set or a system. P and Q
can be sets or relations etc. but P/Q is not a set or class or
function etc. That is the ultimate resolution of all paradoxes (each
created by substitution as I illustrated earlier.)
C-B
Zuhair
largely because of the existence of paradoxes that arise
[/quote]
Charlie+Boo: when the principle is ...... E$#% ('&)( PP¡C10)324
6657 8¦9A at (no spam) CB HED!DGF P¦F('&)( PP¡C10) . ... while creating such a
distinction can be used to avoid an explicit first-order syntax, one
loses ..... 9 S trictly spea"8ing, we should writeA at (no spam) ( D 8 HEF
QCBEDGFH ...
Full text of "Inigo Jones And Ben Jonson Being The Life Of
Inigo ...
Then w cb the Poet cannot know a greater vice, he being y* kind
of ...... And (because method is the mother of discipline) I devide my
Paradoxe into theis ...... Let sweet Orpheus 8ing 14j4j MASK OF THE
FOUR SEASONS. unto his well tun'd ..... cannot avoid saying : e We
have inserted Ben'
Paddy field irrigation systems in Myanmar
capacity building (CB) and human resources development
(HRD). ...... The main lesson from the FAO regional modernization
training programme is a paradox: this challenge is .... groundwater
withdrawal (this is to avoid double counting. ...
Full text of "Chopin S Musical Style"
58, in B Minor , 96, 103, 107 Tarantelle, 63, 96 Trio, Op. 8, in
G minor, 1 5 Valses: Op. 1 8, in Eflat, 13, 44 C^. 34 (as a whole], 44
Op. 34, jVb. i, ...
paradox: unexpectedly low nucleotide diversities and lack of
phylogenetic ...... examination of this trait on the phylogeny reveals
that CB is not as widespread ...... avoid some edge types, but the
mechanisms underlying these spatial ... 8ing ang <w sung ing Sink
Bank or 6uuk suak Sit at sat or Bitten Slay slew slain ... But it
seems to be merely a form adopted to avoid the abruptness of a
direct ...... T,)against, beside, paradox, parochial. Peri, round,
about, as, pe meter. ...... exogly-^ ph\cB. I GnoetOB, known, as,
prog'r?oii- 1 cate. j Gonia, ...
"My CB and computer equipment are taking up all the passenger
seats. .... Avoid those tuck zones. . and they are shut inside the
locket of that cell ...... I must embrace many paradoxes. put me
through. Don't leave me alone with him. ...
8ING-SING WATERBUCK— abnormal head. CHAPTER II THE EQUATORIAL
TRENCH HUNTING ..... The doe I got by a little impromptu drive,
killing her with a Paradox ball ..... but we had selected the longer
way round in order to avoid the heavy march ...... is our friend Mr.
C. B. Perceval, Game-ranger of British East Africa, ...
The story was simple: To avoid being sent to a work farm, ...
"What Is This Thing Called Love", for C. B. Cochran's Wake Up and
Dream ( 1929), ...... Living in Hollywood for Harold was a paradox.
While he was regarded as a star, ...
[quote]A pronoun is employed to avoid an improper or tOQ^&e- quent use
of the noun ; as, ...... times or tenses 3 also the rerbs haUf cb-[/quote]
ttrofff praiMe and blame. ..... Sign— JfocL 8ing%dar. Smgfdar, 1. I
had loved. 1. I had been loving. 2. ...... Para, agaiMt; as, Paradox,,
something contrary to common opinion 12. ... |
|
|
| Back to top |
|
|
|
| Charlie-Boo... |
| Posted: Mon Nov 02, 2009 1:47 am |
|
|
|
Guest
|
On Nov 2, 6:23 am, "Jesse F. Hughes" <je... at (no spam) phiwumbda.org> wrote:
[quote]Charlie-Boo <shymath... at (no spam) gmail.com> writes:
"Zuhair"? It sounds like one of those African natives. Do you have
a bone sticking through your nose?
Congrats, Charlie! A new low!
All this time, I've thought you're just a silly, self-aggrandizing
nincompoop. I had no idea that you were also a disgusting son of a
bitch.
[/quote]
I always knew you weren't very perceptive.
LOL Thanks for the laugh.
C-B
[quote]--
Jesse F. Hughes
"This Trojan appears to utilize a function of the Windows Media DRM
designed to enable license delivery scenarios as part of a social
engineering attack." -- MS candidly explains the role of DRM licenses[/quote] |
|
|
| Back to top |
|
|
|
| Charlie-Boo... |
| Posted: Mon Nov 02, 2009 1:48 am |
|
|
|
Guest
|
On Nov 2, 6:27 am, Aatu Koskensilta <aatu.koskensi... at (no spam) uta.fi> wrote:
[quote]"Jesse F. Hughes" <je... at (no spam) phiwumbda.org> writes:
Charlie-Boo <shymath... at (no spam) gmail.com> writes:
"Zuhair"? It sounds like one of those African natives. Do you have
a bone sticking through your nose?
Congrats, Charlie! A new low!
All this time, I've thought you're just a silly, self-aggrandizing
nincompoop. I had no idea that you were also a disgusting son of a
bitch.
A few years ago Charlie imparted this piece of keen wisdom:
Personal affronts are a waste of time, have no place in a dignified
scholarly discussion such as this, and will make your dick fall off.
[/quote]
Since when was this either dignified or scholarly?
C-B
[quote]--
Aatu Koskensilta (aatu.koskensi... at (no spam) uta.fi)
"Wovon mann nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus[/quote] |
|
|
| Back to top |
|
|
|
| Charlie-Boo... |
| Posted: Mon Nov 02, 2009 2:03 am |
|
|
|
Guest
|
On Nov 2, 6:27 am, Aatu Koskensilta <aatu.koskensi... at (no spam) uta.fi> wrote:
[quote]"Jesse F. Hughes" <je... at (no spam) phiwumbda.org> writes:
Charlie-Boo <shymath... at (no spam) gmail.com> writes:
"Zuhair"? It sounds like one of those African natives. Do you have
a bone sticking through your nose?
Congrats, Charlie! A new low!
All this time, I've thought you're just a silly, self-aggrandizing
nincompoop. I had no idea that you were also a disgusting son of a
bitch.
A few years ago Charlie imparted this piece of keen wisdom:
Personal affronts are a waste of time, have no place in a dignified
scholarly discussion such as this, and will make your dick fall off.
[/quote]
A few minutes ago Atta posted his latest insight into Logic:
"Did you ever actually /read/ (in the ordinary sense of the word) any
of the 300 books
you say you have?"
This place is an utter waste of time. For the life of me I don't know
why you (or occasionally I) waste time with psychotic (believing
statements "without a solid basis") and abusive people - or why you
changed lately from someone with a bit of sense to a look-alike for
the rest of the misfits who vent their insanity here.
My 1st. book, "Handbook of Efficiency Techniques", was very popular
and widely acclaimed. Except for a couple of people and guess what?
They were also people who like programming puzzles, writing articles
about programming and puzzles, and were contract computer
programmers. The people with the same goals as me were the only
critics!
Here the rule is GODLINESS = PROFESSORSHIP
See me on FOM and watch for "Axiomatization of Computer Science" in
your bookstores.
Charlie <= My real name - secret revealed at last.
[quote]--
Aatu Koskensilta (aatu.koskensi... at (no spam) uta.fi)
"Wovon mann nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus[/quote] |
|
|
| Back to top |
|
|
|
| Charlie-Boo... |
| Posted: Mon Nov 02, 2009 2:43 am |
|
|
|
Guest
|
|
| Back to top |
|
|
|
|
|
All times are GMT - 5 Hours
The time now is Sat Nov 21, 2009 9:31 pm
|
|