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 ) ).
On Oct 31, 1:16 pm, zuhair <zaljo...@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.
> 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 ) ).
> 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
No there is axiom 5)
Axiom 5) V is transitive.
i.e. any member of any member of v is a member of V.
On Oct 31, 1:41 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> On Oct 31, 1:16 pm, zuhair <zaljo...@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.
> > 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 ) ).
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 ).
> 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.
On Oct 31, 3:02 pm, zuhair <zaljo...@yahoo.com> wrote:
> On Oct 31, 1:41 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> > On Oct 31, 1:16 pm, zuhair <zaljo...@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.
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.))
> On Oct 31, 2:33 pm, zuhair <zaljo...@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
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.
> 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.
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
> Zuhair
"Zuhair"? It sounds like one of those African natives. Do you have a bone sticking through your nose?
On Oct 31, 12:16 pm, zuhair <zaljo...@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.)
On Nov 1, 5:12 am, Charlie-Boo <shymath...@gmail.com> wrote:
> On Oct 31, 12:16 pm, zuhair <zaljo...@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
Charlie+Boo: when the principle is ...... E$#% ('&)( PP¡C10)324
6657 8¦9A@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@( 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, ... >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- 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. ...
Charlie-Boo <shymath...@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.
-- 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
On Nov 2, 6:23 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> Charlie-Boo <shymath...@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.
> -- > 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
> >> "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.
Since when was this either dignified or scholarly?
> >> "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.
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.
On Nov 2, 6:23 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> Charlie-Boo <shymath...@gmail.com> writes: > > "Zuhair"? It sounds like one of those African natives. Do you have > > a bone sticking through your nose?
> 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.
> -- > 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
On Nov 2, 4:43 am, Charlie-Boo <shymath...@gmail.com> wrote:
> On Nov 2, 6:23 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > Charlie-Boo <shymath...@gmail.com> writes: > > > "Zuhair"? It sounds like one of those African natives. Do you have > > > a bone sticking through your nose?
On Nov 2, 9:35 am, Marshall <marshall.spi...@gmail.com> wrote:
> On Nov 2, 3:23 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > 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.
> Well put.
Yes, it clearly illustrates his lack of knowledge or understanding of Logic, preferring instead to dabble in ad hominem attacks.
On Nov 2, 9:33 am, Marshall <marshall.spi...@gmail.com> wrote:
> On Nov 2, 4:43 am, Charlie-Boo <shymath...@gmail.com> wrote:
> > On Nov 2, 6:23 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > > Charlie-Boo <shymath...@gmail.com> writes: > > > > "Zuhair"? It sounds like one of those African natives. Do you have > > > > a bone sticking through your nose?
In article <5fd6684b-8bd1-4cf4-9da5-994ea21ef...@k4g2000yqb.googlegroups.com>,
Charlie-Boo <shymath...@gmail.com> wrote: >On Nov 2, 9:35=A0am, Marshall <marshall.spi...@gmail.com> wrote: >> On Nov 2, 3:23=A0am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> > All this time, I've thought you're just a silly, self-aggrandizing >> > nincompoop. =A0I had no idea that you were also a disgusting son of a >> > bitch.
>> Well put.
>Yes, it clearly illustrates his lack of knowledge or understanding of >Logic, preferring instead to dabble in ad hominem attacks.
It's not ad hominem to point out that you are a dickhead. Claiming that you are wrong *because* you are a dickhead would be ad hominem. Fortunatly (?), your arguments fail on their own merits; there is no reason to drag your personal failings into it.
> In article <5fd6684b-8bd1-4cf4-9da5-994ea21ef...@k4g2000yqb.googlegroups.com>,
> Charlie-Boo <shymath...@gmail.com> wrote: > >On Nov 2, 9:35=A0am, Marshall <marshall.spi...@gmail.com> wrote: > >> On Nov 2, 3:23=A0am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> >> > All this time, I've thought you're just a silly, self-aggrandizing > >> > nincompoop. =A0I had no idea that you were also a disgusting son of a > >> > bitch.
> >> Well put.
> >Yes, it clearly illustrates his lack of knowledge or understanding of > >Logic, preferring instead to dabble in ad hominem attacks.
> It's not ad hominem to point out that you are a dickhead.
Liar. Ad hominem means "against the man" and a dick(head) is part of a man.
> Claiming that you are wrong *because* you are a dickhead would be ad hominem.
No it wouldn't. It would be attacking the wrong answer, though with specious logic. It would be a non sequitar.
> Fortunatly (?), your arguments fail on their own merits; there is > no reason to drag your personal failings into it.
You're a stupid fuck like the rest of the snarling schizoids.
Charlie-Boo <shymath...@gmail.com> writes: > My 1st. book, "Handbook of Efficiency Techniques", was very popular > and widely acclaimed.
I wonder why you changed the title. Your only published book[1] is "The MUMPS Handbook of Efficiency Techniques: 125 Ways to Make Your Mumps Applications Run Faster."
Rather less impressive sounding, ain't it? No doubt very popular nonetheless.
Footnotes: [1] Correct me if I'm wrong. You keep referring to it as your first book, so surely there's a second book?
-- Jesse F. Hughes -- A lesson in meta-honesty -- Baba: Thanks for being honest. Quincy (age 7): I won't be honest next time. And that's more honesty.
Charlie-Boo <shymath...@gmail.com> writes: > On Nov 2, 9:35 am, Marshall <marshall.spi...@gmail.com> wrote: >> On Nov 2, 3:23 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> > 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.
>> Well put.
> Yes, it clearly illustrates his lack of knowledge or understanding of > Logic, preferring instead to dabble in ad hominem attacks.
No ad hominem, since I was refuting no argument at all. A simple observation: As it happens, Charlie Volkstorf is a disgusting son of a bitch and sufficient evidence of this fact lies four posts up.
Perhaps you should learn what the ad hominem fallacy is. Hint: not every insult is a logical fallacy.
-- God made the bees And the bees make honey. The miller's man does all the work, But the miller makes the money. --- Mother Goose
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