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From: zuhair on 13 Feb 2010 09:21
The following is a theory in FOL with identity "=" membership "e" and
small set-hood "V" , with a universal set in it.

Define: x is small <-> x e V

Extensionality: as in Z

Union, Power, Infinity, Replacement (strong version) All relativised
to V.

Axiom of small set existence: Exist x: x e V

Anti-foundation axiom:

For every z e V Exist x e V For all y ( y e x <-> ( y e z or y=x ) )

Define the i-th union of x by recursion

Ui(y) = y if i=0
Ui(y) = U( Ui-1(y)) if i is a finite ordinal other than 0.

Axiom schema of universal comprehension:

If Phi is a formula in which x is not free ,then all closures of

~ For all y ( (y e V & Phi(y)) -> Exist i ( ~ y e Ui(y) ) )
->
Exist x for all y ( y e x <-> Phi(y) ).

is an axiom.

Theory definition finished.


Now all the paradoxes of Russell's, Buralit-Forti, Cantor's, and
Lesniewski's (the singletons paradox) all would be avoided in this
theory, Even Holmes paradox would be avoided here.

I don't know really if this theory is consistent or not, but anyway it
is so limited to provide significant information other than that ZF
provides.

This theory do prove the existence of a universal set of all sets,
also the set of all sets that are in themselves , also it does prove
the existence of Frege-Russell cardinals for every set in it.

All work that can be done in ZF, can be done in this theory too.

Zuhair







From: zuhair on 13 Feb 2010 14:50
On Feb 13, 9:21 am, zuhair <zaljo...(a)gmail.com> wrote:
> The following is a theory in FOL with identity "=" membership "e" and
> small set-hood "V" , with a universal set in it.
>
> Define: x is small <-> x e V
>
> Extensionality: as in Z
>
> Union, Power, Infinity, Replacement (strong version)  All relativised
> to V.
>
> Axiom of small set existence: Exist x: x e V
>
> Anti-foundation axiom:
>
> For every z e V Exist x e V For all y ( y e x <-> ( y e z or y=x ) )
>
> Define the i-th union of x by recursion

sorry ... of y.

zuhair
>
> Ui(y) = y  if i=0
> Ui(y) = U( Ui-1(y)) if i is a finite ordinal other than 0.
>
> Axiom schema of universal comprehension:
>
> If Phi is a formula in which x is not free ,then all closures of
>
> ~ For all y ( (y e V & Phi(y)) -> Exist i ( ~ y e Ui(y) ) )
> ->
> Exist x for all y ( y e x <-> Phi(y) ).
>
> is an axiom.
>
> Theory definition finished.
>
> Now all the paradoxes of Russell's, Buralit-Forti, Cantor's, and
> Lesniewski's (the singletons paradox) all would be avoided in this
> theory, Even Holmes paradox would be avoided here.
>
> I don't know really if this theory is consistent or not, but anyway it
> is so limited to provide significant information other than that ZF
> provides.
>
> This theory do prove the existence of a universal set of all sets,
> also the set of all sets that are in themselves , also it does prove
> the existence of Frege-Russell cardinals for every set in it.
>
> All work that can be done in ZF, can be done in this theory too.
>
> Zuhair

From: Rupert on 13 Feb 2010 20:42
On Feb 14, 1:21 am, zuhair <zaljo...(a)gmail.com> wrote:
> The following is a theory in FOL with identity "=" membership "e" and
> small set-hood "V" , with a universal set in it.
>
> Define: x is small <-> x e V
>
> Extensionality: as in Z
>
> Union, Power, Infinity, Replacement (strong version)  All relativised
> to V.
>
> Axiom of small set existence: Exist x: x e V
>
> Anti-foundation axiom:
>
> For every z e V Exist x e V For all y ( y e x <-> ( y e z or y=x ) )
>
> Define the i-th union of x by recursion
>
> Ui(y) = y  if i=0
> Ui(y) = U( Ui-1(y)) if i is a finite ordinal other than 0.
>
> Axiom schema of universal comprehension:
>
> If Phi is a formula in which x is not free ,then all closures of
>
> ~ For all y ( (y e V & Phi(y)) -> Exist i ( ~ y e Ui(y) ) )
> ->
> Exist x for all y ( y e x <-> Phi(y) ).
>
> is an axiom.
>
> Theory definition finished.
>
> Now all the paradoxes of Russell's, Buralit-Forti, Cantor's, and
> Lesniewski's (the singletons paradox) all would be avoided in this
> theory, Even Holmes paradox would be avoided here.
>
> I don't know really if this theory is consistent or not, but anyway it
> is so limited to provide significant information other than that ZF
> provides.
>
> This theory do prove the existence of a universal set of all sets,
> also the set of all sets that are in themselves , also it does prove
> the existence of Frege-Russell cardinals for every set in it.
>
> All work that can be done in ZF, can be done in this theory too.
>
> Zuhair

It looks as though you're right about that, yes, it seems clear that
ZF can be interpreted in your theory.

But the question arises why is the theory of interest to study, and is
it consistent.

Can you derive any interesting results in the theory?
From: zuhair on 13 Feb 2010 23:09
On Feb 13, 8:42 pm, Rupert <rupertmccal...(a)yahoo.com> wrote:
> On Feb 14, 1:21 am, zuhair <zaljo...(a)gmail.com> wrote:
>
>
>
>
>
> > The following is a theory in FOL with identity "=" membership "e" and
> > small set-hood "V" , with a universal set in it.
>
> > Define: x is small <-> x e V
>
> > Extensionality: as in Z
>
> > Union, Power, Infinity, Replacement (strong version)  All relativised
> > to V.
>
> > Axiom of small set existence: Exist x: x e V
>
> > Anti-foundation axiom:
>
> > For every z e V Exist x e V For all y ( y e x <-> ( y e z or y=x ) )
>
> > Define the i-th union of x by recursion
>
> > Ui(y) = y  if i=0
> > Ui(y) = U( Ui-1(y)) if i is a finite ordinal other than 0.
>
> > Axiom schema of universal comprehension:
>
> > If Phi is a formula in which x is not free ,then all closures of
>
> > ~ For all y ( (y e V & Phi(y)) -> Exist i ( ~ y e Ui(y) ) )
> > ->
> > Exist x for all y ( y e x <-> Phi(y) ).
>
> > is an axiom.
>
> > Theory definition finished.
>
> > Now all the paradoxes of Russell's, Buralit-Forti, Cantor's, and
> > Lesniewski's (the singletons paradox) all would be avoided in this
> > theory, Even Holmes paradox would be avoided here.
>
> > I don't know really if this theory is consistent or not, but anyway it
> > is so limited to provide significant information other than that ZF
> > provides.
>
> > This theory do prove the existence of a universal set of all sets,
> > also the set of all sets that are in themselves , also it does prove
> > the existence of Frege-Russell cardinals for every set in it.
>
> > All work that can be done in ZF, can be done in this theory too.
>
> > Zuhair
>
> It looks as though you're right about that, yes, it seems clear that
> ZF can be interpreted in your theory.
>
> But the question arises why is the theory of interest to study, and is
> it consistent.
>
> Can you derive any interesting results in the theory?

I already mentioned that this theory have only scanty results beyond
those of ZF, that is if it turns to be consistent.

I am not sure if interesting results can be derived in this theory, I
only presented it because I saw it overcome all the paradoxes that I
know which are usually mentioned with set theories, also it can prove
the existence of the set of all sets, also the set of all sets that
are in themselves, it can also prove the existence of pairing of any
set with a set that is in itself, also the union any set with a set
that is in itself, but all that material doesn't seem to be
interesting really, and perhaps some modified form of separation, and
even modified form of power, Frege-Russell's cardinals are also
defined in this theory.

My primary interest here is to show that such a theory can avoid the
known paradoxes while at the same time having universal sets in it.

This theory doesn't mention any stratification on formulas in it, and
it doesn't relay on this methodology at all, unlike NF and related
theories.

Avoiding Paradoxes:

(1) Russell's: Let's take the formula ~ y e y.

Now substitute this in universal comprehension and we get:

~ For all y ( (y e V & ~ y e y) -> Exist i ( ~ y e Ui(y) ) )
->
Exist x for all y ( y e x <-> Phi(y) ).

Now lets see if the antecedent is true or not?

Let's take i=0

so the antecedent would become:

~ For all y ( (y e V & ~ y e y) -> ~ y e y ) )

which is clearly false. Thus we cannot have

Exist x for all y ( y e x <-> ~ y e y ).

Thus avoiding the paradox! QED

(2) Burali-Forti:

Let Phi(y) <-> y is ordinal

Now lets substitute in universal comprehension

so we'll have:

~ For all y ( (y e V & y is ordinal) -> Exist i ( ~ y e Ui(y) ) )
->
Exist x for all y ( y e x <-> Phi(y) ).

Let i be any finite ordinal, and it is clear that the antecedent would
be false, thus we cannot have the left hand of the implication above,
thus avoiding the paradox.

(3) Cantor's paradox: We can have the set of all sets, but since there
is no separation here over such set, then actually Cantor's paradox
disappear.

(4) Singletons Paradox: We cannot have a set of all singletons that
are not in their sole members, since this would render the antecedent
of universal comprehension false. Even if we take the set of all
doubletons that are in themselves and not in their other member (the
non trivial member), still this set itself is not in itself and thus
do not have a doubleton set that is in itself and having it as a
member, so this paradox disappear.

(5) Holmes paradox, it depends on having a function between the
universal sets and the set of all natural numbers, which is not
necessarily the case here.

So all known paradoxes are avoided with this theory.

However weather it is consistent or not, that is another question
really, but my guess is that it is not.

Zuhair



From: zuhair on 14 Feb 2010 14:48
On Feb 13, 11:09 pm, zuhair <zaljo...(a)gmail.com> wrote:
> On Feb 13, 8:42 pm, Rupert <rupertmccal...(a)yahoo.com> wrote:
>
>
>
>
>
> > On Feb 14, 1:21 am, zuhair <zaljo...(a)gmail.com> wrote:
>
> > > The following is a theory in FOL with identity "=" membership "e" and
> > > small set-hood "V" , with a universal set in it.
>
> > > Define: x is small <-> x e V
>
> > > Extensionality: as in Z
>
> > > Union, Power, Infinity, Replacement (strong version)  All relativised
> > > to V.
>
> > > Axiom of small set existence: Exist x: x e V
>
> > > Anti-foundation axiom:
>
> > > For every z e V Exist x e V For all y ( y e x <-> ( y e z or y=x ) )
>
> > > Define the i-th union of x by recursion
>
> > > Ui(y) = y  if i=0
> > > Ui(y) = U( Ui-1(y)) if i is a finite ordinal other than 0.
>
> > > Axiom schema of universal comprehension:
>
> > > If Phi is a formula in which x is not free ,then all closures of
>
> > > ~ For all y ( (y e V & Phi(y)) -> Exist i ( ~ y e Ui(y) ) )
> > > ->
> > > Exist x for all y ( y e x <-> Phi(y) ).
>
> > > is an axiom.
>
> > > Theory definition finished.
>
> > > Now all the paradoxes of Russell's, Buralit-Forti, Cantor's, and
> > > Lesniewski's (the singletons paradox) all would be avoided in this
> > > theory, Even Holmes paradox would be avoided here.
>
> > > I don't know really if this theory is consistent or not, but anyway it
> > > is so limited to provide significant information other than that ZF
> > > provides.
>
> > > This theory do prove the existence of a universal set of all sets,
> > > also the set of all sets that are in themselves , also it does prove
> > > the existence of Frege-Russell cardinals for every set in it.
>
> > > All work that can be done in ZF, can be done in this theory too.
>
> > > Zuhair
>
> > It looks as though you're right about that, yes, it seems clear that
> > ZF can be interpreted in your theory.
>
> > But the question arises why is the theory of interest to study, and is
> > it consistent.
>
> > Can you derive any interesting results in the theory?
>
> I already mentioned that this theory have only scanty results beyond
> those of ZF, that is if it turns to be consistent.
>
> I am not sure if interesting results can be derived in this theory, I
> only presented it because I saw it overcome all the paradoxes that I
> know which are usually mentioned with set theories, also it can prove
> the existence of the set of all sets, also the set of all sets that
> are in themselves, it can also prove the existence of pairing of any
> set with a set that is in itself, also the union any set with a set
> that is in itself, but all that material doesn't seem to be
> interesting really, and perhaps some modified form of separation, and
> even modified form of power, Frege-Russell's cardinals are also
> defined in this theory.
>
> My primary interest here is to show that such a theory can avoid the
> known paradoxes while at the same time having universal sets in it.
>
> This theory doesn't mention any stratification on formulas in it, and
> it doesn't relay on this methodology at all, unlike NF and related
> theories.
>
> Avoiding Paradoxes:
>
> (1) Russell's: Let's take the formula ~ y e y.
>
> Now substitute this in universal comprehension and we get:
>
> ~ For all y ( (y e V & ~ y e y) -> Exist i ( ~ y e Ui(y) ) )
> ->
> Exist x for all y ( y e x <-> Phi(y) ).
>
> Now lets see if the antecedent is true or not?
>
> Let's take i=0
>
> so the antecedent would become:
>
> ~ For all y ( (y e V & ~ y e y) -> ~ y e y ) )
>
> which is clearly false. Thus we cannot have
>
> Exist x for all y ( y e x <-> ~ y e y ).
>
> Thus avoiding the paradox! QED
>
> (2) Burali-Forti:
>
> Let Phi(y) <-> y is ordinal
>
> Now lets substitute in universal comprehension
>
> so we'll have:
>
> ~ For all y ( (y e V & y is ordinal) -> Exist i ( ~ y e Ui(y) ) )
> ->
> Exist x for all y ( y e x <-> Phi(y) ).
>
> Let i be any finite ordinal, and it is clear that the antecedent would
> be false, thus we cannot have the left hand of the implication above,
> thus avoiding the paradox.
>
> (3) Cantor's paradox: We can have the set of all sets, but since there
> is no separation here over such set, then actually Cantor's paradox
> disappear.
>
> (4) Singletons Paradox: We cannot have a set of all singletons that
> are not in their sole members, since this would render the antecedent
> of universal comprehension false. Even if we take the set of all
> doubletons that are in themselves and not in their other member (the
> non trivial member), still this set itself is not in itself and thus
> do not have a doubleton set that is in itself and having it as a
> member, so this paradox disappear.
>
> (5) Holmes paradox, it depends on having a function between the
> universal sets and the set of all natural numbers, which is not
> necessarily the case here.
>

On second look, I think this theory might be inconsistent, since it
appears that Holmes paradox can be reproduced in it.

Zuhair


> So all known paradoxes are avoided with this theory.
>
> However weather it is consistent or not, that is another question
> really, but my guess is that it is not.
>
> Zuhair

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