Prev: Hemant Pandey from India Solves P vs. NP
Next: "A" defining Finite Number for High School students #257; 2nd ed; Correcting Math
From: zuhair on 31 Oct 2009 15:33
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




From: zuhair on 31 Oct 2009 16:10
On Oct 31, 2:33 pm, zuhair <zaljo...(a)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
From: zuhair on 31 Oct 2009 17:08
On Oct 31, 3:10 pm, zuhair <zaljo...(a)yahoo.com> wrote:
> On Oct 31, 2:33 pm, zuhair <zaljo...(a)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.

Zuhair
From: Charlie-Boo on 1 Nov 2009 07:02

> > 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.

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?
From: Marshall on 2 Nov 2009 09:33
On Nov 2, 4:43 am, Charlie-Boo <shymath...(a)gmail.com> wrote:
> On Nov 2, 6:23 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
> > Charlie-Boo <shymath...(a)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!
>
> Low?  What is low about African culture?
>
> http://images.google.com/imgres?imgurl=http://farm3.static.flickr.com....

Unfortunately for your argument Papua New Guinea is not in Africa.


Marshall
 |  Next  |  Last
Pages: 1 2 3 4 5 6
Prev: Hemant Pandey from India Solves P vs. NP
Next: "A" defining Finite Number for High School students #257; 2nd ed; Correcting Math