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From: zuhair on 15 Feb 2010 00:08
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, 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: William Elliot on 15 Feb 2010 00:36
On Sun, 14 Feb 2010, zuhair wrote:

> 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.
>
> I think these formulas are not suitable for constructing sets, and
> they'd better be avoided.

That's what Quine did in New Foundations (NF) to disallow unstratified
propositions from forming sets. NF -> ~AxC.

> 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.
>
Is self comparing the same as unstratified?

> 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!

IIRC, the quest for proving consistency of NF continues.

Conjecture. If one of the set theories, NF, Z, NBG is proven
consistent, then the others are consistent also.

> 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

A space is to follow each of those commas.
Otherwise the line gets too tedious to read.

> 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.
>
Here's my notes on NF.

-- New Foundations Home Page; math.boisestate.edu/~holmes/holmes/nf.html
axioms of Quine's New Foundations (NF)
* Stratified comprehension is an axiom scheme, replaceable by finite
many of its instances. Using the finite axiomatization removes
the need for stratification. Hailperin, T. [1944]
A set of axioms for logic. Journal of Symbolic Logic 9, pp. 1-19.
* NF disproves the Axiom of Choice. Specker, E.P. [1953]
The axiom of choice in Quine's new foundations for mathematical logic.
Proceedings of the National Academy of Sciences of the USA 39, pp. 972-975.

Axiom Scheme of Stratified Comprehension:
If psi is a stratified formula in which the variable A is not free,
(exists A.(forall x.x in A <=> psi)).

Definition: A function s from variables to integers is a stratification
of psi when for each atomic subformula 'x in y' of psi, s(`x')+1 = s(`y').
(If equality present as a primitive, then for each atomic subformula `x =
y' in psi, s(`x') = s(`y').) A formula psi is stratified when there's a
stratification of psi.

Definition: A stratification of the abstract { x | psi } is defined
as a stratification of the formula (forall x.x in A <=> psi).

Definition: x = y is an abbreviation for (forall z.x in z <=> y in z).
Axiom of equality: (forall x,y,z. x = y => (x in z <=> y in z))

----
From: zuhair on 15 Feb 2010 01:02
On Feb 15, 12:36 am, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Sun, 14 Feb 2010, zuhair wrote:
> > 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.
>
> > I think these formulas are not suitable for constructing sets, and
> > they'd better be avoided.
>
> That's what Quine did in New Foundations (NF) to disallow unstratified
> propositions from forming sets.  NF -> ~AxC.
>
> > 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.
>
> Is self comparing the same as unstratified?

No, they are not the same, an obvious example is the formula y=y
this is self comparing formula but it is stratified!
>
>
>
> > 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.

Note: "non comparing" here refers to "non self comparing" as
characterized above.
>
> > Theory definition finished/
>
> > My personal guess is that this theory must be inconsistent!
>
> IIRC, the quest for proving consistency of NF continues.
>
> Conjecture.  If one of the set theories, NF, Z, NBG is proven
> consistent, then the others are consistent also.
>
> > 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
>
> A space is to follow each of those commas.
> Otherwise the line gets too tedious to read.
>
> > 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.
>
> Here's my notes on NF.
>
> -- New Foundations Home Page;  math.boisestate.edu/~holmes/holmes/nf.html
> axioms of Quine's New Foundations (NF)
>    * Stratified comprehension is an axiom scheme, replaceable by finite
>         many of its instances.  Using the finite axiomatization removes
>         the need for stratification.  Hailperin, T. [1944]
>         A set of axioms for logic.  Journal of Symbolic Logic 9, pp. 1-19.
>    * NF disproves the Axiom of Choice.  Specker, E.P. [1953]
> The axiom of choice in Quine's new foundations for mathematical logic.
> Proceedings of the National Academy of Sciences of the USA 39, pp. 972-975.
>
> Axiom Scheme of Stratified Comprehension:
> If psi is a stratified formula in which the variable A is not free,
>         (exists A.(forall x.x in A <=> psi)).
>
> Definition:  A function s from variables to integers is a stratification
> of psi when for each atomic subformula 'x in y' of psi, s(`x')+1 = s(`y').
> (If equality present as a primitive, then for each atomic subformula `x =
> y' in psi, s(`x') = s(`y').)  A formula psi is stratified when there's a
> stratification of psi.
>
> Definition:  A stratification of the abstract { x | psi } is defined
>         as a stratification of the formula (forall x.x in A <=> psi).
>
> Definition:  x = y is an abbreviation for (forall z.x in z <=> y in z).
> Axiom of equality:  (forall x,y,z. x = y => (x in z <=> y in z))
>
> ----

From: zuhair on 15 Feb 2010 01:53
On Feb 15, 12:36 am, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Sun, 14 Feb 2010, zuhair wrote:
> > 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.
>
> > I think these formulas are not suitable for constructing sets, and
> > they'd better be avoided.
>
> That's what Quine did in New Foundations (NF) to disallow unstratified
> propositions from forming sets.  NF -> ~AxC.
>
> > 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.
>
> Is self comparing the same as unstratified?

No.

You can have a self comparing formula that is stratified, for example:

y=y

You can have a self comparing formula that is unstratified, for
example:

Exist a ( y e a or y=a )

You can have a non self comparing formula that is stratified, for
example:

for all z (z e y -> z e a)

You can have a non self comparing formula that is unstratified, for
example:

y e a or y=a

So "self comparison" and "stratification" are separate concepts.

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!
>
> IIRC, the quest for proving consistency of NF continues.
>
> Conjecture.  If one of the set theories, NF, Z, NBG is proven
> consistent, then the others are consistent also.
>
> > 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
>
> A space is to follow each of those commas.
> Otherwise the line gets too tedious to read.
>
> > 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.
>
> Here's my notes on NF.
>
> -- New Foundations Home Page;  math.boisestate.edu/~holmes/holmes/nf.html
> axioms of Quine's New Foundations (NF)
>    * Stratified comprehension is an axiom scheme, replaceable by finite
>         many of its instances.  Using the finite axiomatization removes
>         the need for stratification.  Hailperin, T. [1944]
>         A set of axioms for logic.  Journal of Symbolic Logic 9, pp. 1-19.
>    * NF disproves the Axiom of Choice.  Specker, E.P. [1953]
> The axiom of choice in Quine's new foundations for mathematical logic.
> Proceedings of the National Academy of Sciences of the USA 39, pp. 972-975.
>
> Axiom Scheme of Stratified Comprehension:
> If psi is a stratified formula in which the variable A is not free,
>         (exists A.(forall x.x in A <=> psi)).
>
> Definition:  A function s from variables to integers is a stratification
> of psi when for each atomic subformula 'x in y' of psi, s(`x')+1 = s(`y').
> (If equality present as a primitive, then for each atomic subformula `x =
> y' in psi, s(`x') = s(`y').)  A formula psi is stratified when there's a
> stratification of psi.
>
> Definition:  A stratification of the abstract { x | psi } is defined
>         as a stratification of the formula (forall x.x in A <=> psi).
>
> Definition:  x = y is an abbreviation for (forall z.x in z <=> y in z).
> Axiom of equality:  (forall x,y,z. x = y => (x in z <=> y in z))
>
> ----

From: zuhair on 15 Feb 2010 02:14
On Feb 15, 12:36 am, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Sun, 14 Feb 2010, zuhair wrote:
> > 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.
>
> > I think these formulas are not suitable for constructing sets, and
> > they'd better be avoided.
>
> That's what Quine did in New Foundations (NF) to disallow unstratified
> propositions from forming sets.  NF -> ~AxC.
>
> > 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.
>
> Is self comparing the same as unstratified?
>
>
>
> > 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!
>
> IIRC, the quest for proving consistency of NF continues.
>
> Conjecture.  If one of the set theories, NF, Z, NBG is proven
> consistent, then the others are consistent also.

No that is not correct, NF is not known to be equi-consistent with Z,
NFU is a different matter and it is known to be equi-consistent with
the other theories, but not NF.
>
> > 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
>
> A space is to follow each of those commas.
> Otherwise the line gets too tedious to read.
>
> > 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.
>
> Here's my notes on NF.
>
> -- New Foundations Home Page;  math.boisestate.edu/~holmes/holmes/nf.html
> axioms of Quine's New Foundations (NF)
>    * Stratified comprehension is an axiom scheme, replaceable by finite
>         many of its instances.  Using the finite axiomatization removes
>         the need for stratification.  Hailperin, T. [1944]
>         A set of axioms for logic.  Journal of Symbolic Logic 9, pp. 1-19.
>    * NF disproves the Axiom of Choice.  Specker, E.P. [1953]
> The axiom of choice in Quine's new foundations for mathematical logic.
> Proceedings of the National Academy of Sciences of the USA 39, pp. 972-975.
>
> Axiom Scheme of Stratified Comprehension:
> If psi is a stratified formula in which the variable A is not free,
>         (exists A.(forall x.x in A <=> psi)).
>
> Definition:  A function s from variables to integers is a stratification
> of psi when for each atomic subformula 'x in y' of psi, s(`x')+1 = s(`y').
> (If equality present as a primitive, then for each atomic subformula `x =
> y' in psi, s(`x') = s(`y').)  A formula psi is stratified when there's a
> stratification of psi.
>
> Definition:  A stratification of the abstract { x | psi } is defined
>         as a stratification of the formula (forall x.x in A <=> psi).
>
> Definition:  x = y is an abbreviation for (forall z.x in z <=> y in z).
> Axiom of equality:  (forall x,y,z. x = y => (x in z <=> y in z))
>
> ----

It is true that finite axiomatization of NFU, would remove the need
for stratification, but this is not the full picture, actually even in
the finite version of NFU, stratification would be proved as a
theorem, and most interesting results from it are actually derived
through stratification! so still stratification is the main tool of
inference even in the finite versions of NFU, only the axioms of the
finite version of NFU don't need stratification, but most work done in
these versions actually relies heavily on stratification.

Zuhair

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