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From: zuhair on 15 Feb 2010 17:19
On Feb 15, 12:08 am, zuhair <zaljo...(a)gmail.com> wrote:
> Hi all,
>
> The paradoxes of set theory: Russell's, Burali-Forti's,
> Cantor's, Lesniewski's, and even Holmes paradox,
>
> I noticed that all such paradoxes occur with formulas that are
> binary relations having One free variable in them,i.e formulas in
> which a free variable occur in two places having no other free
> variables in it, so either the formula is atomic with ONE free
> variable and a primitive binary relation, or the formula having ONE
> free variable occurring twice and the other variables bound,
> for example
>
> y=y
> y e y
> ~ y e y
> Exist x ( x e y & ~ y e x )
> Exist x,z ( y e x & x e z & z e y )
> Exist x ( x e y & ~ Exist c ( c e x & c e y ) )
>
> all such formulas are binary relation formulas of y with itself , i.e.
> of
> the general form y R y. So they are formulas that compare y with
> itself.
>
> It is not a surprise that such formulas would produce paradoxes,
> since the class that is defined after such formulas might have this
> relation with itself.
>
> I think these formulas are not suitable for constructing sets, and
> they'd better be avoided.
>
> So any formula Phi(y) having a sub-formula that is a binary relation
> of One free variable with itself  is to be named as
> a " self comparing formula", and it is to be avoided, so
> Phi(y) is non self comparing formula if non of its sub-formulas is
> self comparing formula.

Correction:

Phi(y) is non self comparing formula if non of its sub-formulas is
a binary relation of One free variable with itself.

Zuhair

>
> So we may define the following theory T in FOL with identity and e:
>
> T is the set of all sentences entailed (from FOL with identity and
> epsilon membership) by the axiom schema outlined below the following
> definition:
>
> Define (set):
>  x is a set iff for all y ( for all z ( z e y iff z e x) -> y=x )
>  or equivalently:
>  x is a set iff ~ exist y ( ~ y=x & for all z ( z e y iff z e x) )
>
> Axiom Schema of Comprehension: If Phi is a non comparing
> formula in which at least y is free, and in which x is not free, then
> all closures of
>
> Exist a set x for all y ( y e x iff Phi )
>
> are axioms.
>
> Theory definition finished/
>
> My personal guess is that this theory must be inconsistent!
> Theorems of this theory include the existence of a set of all sets,
> the set of all non sets, also the existence of an empty set (use the
> predicate (y=a & ~y=a)), also pairing,union,power,also the set of all
> supersets of a set, all are theorems here, separation using non
> comparing formula, replacement using non comparing formulas.
>
> I don't know if infinity can be proved in this theory? and what is the
> stance of this theory with choice?
>
> Anyhow I do think that this theory is mostly inconsistent, and it
> would be a good exercise to prove its inconsistency.
>
> Zuhair

From: zuhair on 15 Feb 2010 17:26
On Feb 15, 12:08 am, zuhair <zaljo...(a)gmail.com> wrote:
> Hi all,
>
> The paradoxes of set theory: Russell's, Burali-Forti's,
> Cantor's, Lesniewski's, and even Holmes paradox,
>
> I noticed that all such paradoxes occur with formulas that are
> binary relations having One free variable in them,i.e formulas in
> which a free variable occur in two places having no other free
> variables in it, so either the formula is atomic with ONE free
> variable and a primitive binary relation, or the formula having ONE
> free variable occurring twice and the other variables bound,
> for example
>
> y=y
> y e y
> ~ y e y
> Exist x ( x e y & ~ y e x )
> Exist x,z ( y e x & x e z & z e y )
> Exist x ( x e y & ~ Exist c ( c e x & c e y ) )
>
> all such formulas are binary relation formulas of y with itself , i.e.
> of
> the general form y R y. So they are formulas that compare y with
> itself.
>
> It is not a surprise that such formulas would produce paradoxes,
> since the class that is defined after such formulas might have this
> relation with itself.
>
> I think these formulas are not suitable for constructing sets, and
> they'd better be avoided.
>
> So any formula Phi(y) having a sub-formula that is a binary relation
> of One free variable with itself  is to be named as
> a " self comparing formula", and it is to be avoided, so
> Phi(y) is non self comparing formula if non of its sub-formulas is
> self comparing formula.
>
> So we may define the following theory T in FOL with identity and e:
>
> T is the set of all sentences entailed (from FOL with identity and
> epsilon membership) by the axiom schema outlined below the following
> definition:
>
> Define (set):
>  x is a set iff for all y ( for all z ( z e y iff z e x) -> y=x )
>  or equivalently:
>  x is a set iff ~ exist y ( ~ y=x & for all z ( z e y iff z e x) )
>
> Axiom Schema of Comprehension: If Phi is a non comparing
> formula in which at least y is free, and in which x is not free, then
> all closures of
>
> Exist a set x for all y ( y e x iff Phi )
>
> are axioms.
>
> Theory definition finished/
>
> My personal guess is that this theory must be inconsistent!
> Theorems of this theory include the existence of a set of all sets,
> the set of all non sets,

Clarification: the formula x is a set is
a self comparing formula since x appears twice in it
and all other variables in it are bound, so we cannot
have the set of all sets, nor do we have the set of all
non sets using these formulas in comprehension, however
we can have a set of all supersets of any set x, and the
union of this set would be the set of all sets, and I think
in a similar manner we might have the set of all non sets.

Zuhair



also the existence of an empty set (use the
> predicate (y=a & ~y=a)), also pairing,union,power,also the set of all
> supersets of a set, all are theorems here, separation using non
> comparing formula, replacement using non comparing formulas.
>
> I don't know if infinity can be proved in this theory? and what is the
> stance of this theory with choice?
>
> Anyhow I do think that this theory is mostly inconsistent, and it
> would be a good exercise to prove its inconsistency.
>
> Zuhair

From: zuhair on 15 Feb 2010 18:38
On Feb 15, 5:19 pm, zuhair <zaljo...(a)gmail.com> wrote:
> On Feb 15, 12:08 am, zuhair <zaljo...(a)gmail.com> wrote:
>
>
>
>
>
> > Hi all,
>
> > The paradoxes of set theory: Russell's, Burali-Forti's,
> > Cantor's, Lesniewski's, and even Holmes paradox,
>
> > I noticed that all such paradoxes occur with formulas that are
> > binary relations having One free variable in them,i.e formulas in
> > which a free variable occur in two places having no other free
> > variables in it, so either the formula is atomic with ONE free
> > variable and a primitive binary relation, or the formula having ONE
> > free variable occurring twice and the other variables bound,
> > for example
>
> > y=y
> > y e y
> > ~ y e y
> > Exist x ( x e y & ~ y e x )
> > Exist x,z ( y e x & x e z & z e y )
> > Exist x ( x e y & ~ Exist c ( c e x & c e y ) )
>
> > all such formulas are binary relation formulas of y with itself , i.e.
> > of
> > the general form y R y. So they are formulas that compare y with
> > itself.
>
> > It is not a surprise that such formulas would produce paradoxes,
> > since the class that is defined after such formulas might have this
> > relation with itself.
>
> > I think these formulas are not suitable for constructing sets, and
> > they'd better be avoided.
>
> > So any formula Phi(y) having a sub-formula that is a binary relation
> > of One free variable with itself  is to be named as
> > a " self comparing formula", and it is to be avoided, so
> > Phi(y) is non self comparing formula if non of its sub-formulas is
> > self comparing formula.
>
> Correction:
>
> Phi(y) is non self comparing formula if non of its sub-formulas is
>  a binary relation of One free variable with itself.

Clarification:

Phi(y) is non self comparing formula if non of its sub-formulas is
a formula having one free variable occurring twice in it.

Zuhair


>
> Zuhair
>
>
>
> > So we may define the following theory T in FOL with identity and e:
>
> > T is the set of all sentences entailed (from FOL with identity and
> > epsilon membership) by the axiom schema outlined below the following
> > definition:
>
> > Define (set):
> >  x is a set iff for all y ( for all z ( z e y iff z e x) -> y=x )
> >  or equivalently:
> >  x is a set iff ~ exist y ( ~ y=x & for all z ( z e y iff z e x) )
>
> > Axiom Schema of Comprehension: If Phi is a non comparing
> > formula in which at least y is free, and in which x is not free, then
> > all closures of
>
> > Exist a set x for all y ( y e x iff Phi )
>
> > are axioms.
>
> > Theory definition finished/
>
> > My personal guess is that this theory must be inconsistent!
> > Theorems of this theory include the existence of a set of all sets,
> > the set of all non sets, also the existence of an empty set (use the
> > predicate (y=a & ~y=a)), also pairing,union,power,also the set of all
> > supersets of a set, all are theorems here, separation using non
> > comparing formula, replacement using non comparing formulas.
>
> > I don't know if infinity can be proved in this theory? and what is the
> > stance of this theory with choice?
>
> > Anyhow I do think that this theory is mostly inconsistent, and it
> > would be a good exercise to prove its inconsistency.
>
> > Zuhair

From: Aatu Koskensilta on 15 Feb 2010 18:47
zuhair <zaljohar(a)gmail.com> writes:

> Phi(y) is non self comparing formula if non of its sub-formulas is a
> formula having one free variable occurring twice in it.

In presence of equality every formula is logically equivalent to a
non-self-comparing formula. Just replace

xRx

with

y=x & xRy

and add an existential quantifier to bind y.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 15 Feb 2010 19:58
Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes:

> zuhair <zaljohar(a)gmail.com> writes:
>
>> Phi(y) is non self comparing formula if non of its sub-formulas is a
>> formula having one free variable occurring twice in it.
>
> In presence of equality every formula is logically equivalent to a
> non-self-comparing formula. Just replace
>
> xRx
>
> with
>
> y=x & xRy
>
> and add an existential quantifier to bind y.

Ah, it seems I misread your definition. Apologies for that.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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