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| zuhair... |
 Posted: Thu Nov 05, 2009 1:28 pm |
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Guest
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Hi all,
The following is a set theory in FOL with identity and epsilon
membeship that use the concept of hyperregularity.
E: existential quantifier
A: universal quantifier
Define (regular):
x is regular <-> (Ey(yex) ->Ey(yex & ~Ez(zey& zex)))
Define (hyperregular):
x is hyperregular <-> (x is regular & Ay(yex -> y is regular))
Define (subclass):
x subclass y <-> Az ( zex -> zey )
Define (set):
x is a set <-> Ey,z (y is regular & z is hyperregular &
yez & x subclass y)
Axioms:
1. Extensionality: Az (zex<->zey) ->x=y
2. Set existence: Ex ( x is a set )
3.Class Comprehension: If phi is a formula in which at least y is
free, and in which x is not free, then all closures of
Ex Ay ( yex <-> ( y is a set & phi ) )
are axioms.
Define: x={y|phi} <-> Ay ( yex <-> ( y is a set & phi ) )
4. Set Comprehension: if phi is a formula in which c is not free,
having at least y occurring free, and x1,...,xn are its parameters,
and if Q1,...,Qm are all proper subformulas of phi, then
Ac,x1,...,xn
( (x1,...,xn all are sets &
Ay (phi(y) -> y is a set) & c={y|phi} &
Au1,...,um ((u1={y|Q1}&...& um={y|Qm}) ->
(~u1=c &...&~um=c)))
-> c is a set ).
are axioms.
Theory definition finished/
Now I claim that this set theory have ZF as a sub-theory of it. We can
have all rules of pairing, union, power, infinity, separation and
replacement of "sets" in this theory.
Zuhair |
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| zuhair... |
| Posted: Thu Nov 05, 2009 1:47 pm |
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Guest
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On Nov 5, 6:28 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
[quote]Hi all,
The following is a set theory in FOL with identity and epsilon
membeship that use the concept of hyperregularity.
E: existential quantifier
A: universal quantifier
Define (regular):
x is regular <-> (Ey(yex) ->Ey(yex & ~Ez(zey& zex)))
Define (hyperregular):
x is hyperregular <-> (x is regular & Ay(yex -> y is regular))
Define (subclass):
x subclass y <-> Az ( zex -> zey )
Define (set):
x is a set <-> Ey,z (y is regular & z is hyperregular &
yez & x subclass y)
Axioms:
1. Extensionality: Az (zex<->zey) ->x=y
2. Set existence: Ex ( x is a set )
3.Class Comprehension: If phi is a formula in which at least y is
free, and in which x is not free, then all closures of
Ex Ay ( yex <-> ( y is a set & phi ) )
are axioms.
Define: x={y|phi} <-> Ay ( yex <-> ( y is a set & phi ) )
4. Set Comprehension: if phi is a formula in which c is not free,
having at least y occurring free, and x1,...,xn are its parameters,
and if Q1,...,Qm are all proper subformulas of phi, then
Ac,x1,...,xn
( (x1,...,xn all are sets &
Ay (phi(y) -> y is a set) & c={y|phi} &
Au1,...,um ((u1={y|Q1}&...& um={y|Qm}) -
(~u1=c &...&~um=c)))
-> c is a set ).
are axioms.
Theory definition finished/
Now I claim that this set theory have ZF as a sub-theory of it. We can
have all rules of pairing, union, power, infinity, separation and
replacement of "sets" in this theory.
Zuhair
[/quote]
Though I am not sure, but I think this theory needs axiom of Infinity:
5: Infinity: The class of all finite Von Neumann's ordinals is a set.
Zuhair |
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| zuhair... |
| Posted: Thu Nov 05, 2009 2:46 pm |
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Guest
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On Nov 5, 6:47 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
[quote]On Nov 5, 6:28 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
Hi all,
The following is a set theory in FOL with identity and epsilon
membeship that use the concept of hyperregularity.
E: existential quantifier
A: universal quantifier
Define (regular):
x is regular <-> (Ey(yex) ->Ey(yex & ~Ez(zey& zex)))
Define (hyperregular):
x is hyperregular <-> (x is regular & Ay(yex -> y is regular))
Define (subclass):
x subclass y <-> Az ( zex -> zey )
Define (set):
x is a set <-> Ey,z (y is regular & z is hyperregular &
yez & x subclass y)
Axioms:
1. Extensionality: Az (zex<->zey) ->x=y
2. Set existence: Ex ( x is a set )
3.Class Comprehension: If phi is a formula in which at least y is
free, and in which x is not free, then all closures of
Ex Ay ( yex <-> ( y is a set & phi ) )
are axioms.
Define: x={y|phi} <-> Ay ( yex <-> ( y is a set & phi ) )
4. Set Comprehension: if phi is a formula in which c is not free,
having at least y occurring free, and x1,...,xn are its parameters,
and if Q1,...,Qm are all proper subformulas of phi, then
Ac,x1,...,xn
( (x1,...,xn all are sets &
Ay (phi(y) -> y is a set) & c={y|phi} &
Au1,...,um ((u1={y|Q1}&...& um={y|Qm}) -
(~u1=c &...&~um=c)))
-> c is a set ).
are axioms.
Theory definition finished/
Now I claim that this set theory have ZF as a sub-theory of it. We can
have all rules of pairing, union, power, infinity, separation and
replacement of "sets" in this theory.
Zuhair
Though I am not sure, but I think this theory needs axiom of Infinity:
5: Infinity: The class of all finite Von Neumann's ordinals is a set.
Zuhair
[/quote]
I am not sure of Infinity really.
We can use the formula Az((0ez&Am(mez->mU{m}ez)) -> yez)
an substitute it in class comprehension, then it appears that Infinity
is a theorem of this theory, i.e. no need to axiomatize it, yet I am
not quite sure of this now.
On the other hand if Q is a proper subformula of phi is defined in
such a manner
that for Q to be proper subformula of phi then Q should be a substring
of the symbols in phi that is a formula provided that the remaining
symbols in phi that are not in Q constitute a formula also, then there
will be no need for axiom 2 (which is ad hoc axiom by the way), since
y=y will not be a subformula of ~y=y , since what is left of ~y=y
after taking y=y from it is"~" which is not a formula, so according to
this definition although y=y is a proper substring of ~y=y , yet it is
not a subformula of ~y=y, because ~ is not a formula. while for
example Let the formula phi be
" x is a set & x is ordinal" here "x is a set" is a proper subformula
of phi and
also x is ordinal is also a proper subformula of phi.
Also one can add some restrictions on the definition of a subformula
like saying that a proper subformula should have the principal
variable of the formula and all parameters of the subformula should be
a subset of the parameters of the formula, these are also reasonable
restrictions on the definition of subformulas.
Taking all these restrictions on the definition of "proper
subformula", we'll not need axiom 2, and the theory will consist of
only three axioms schemes, and will not possess any ad-hoc-sens in its
exhibition.
Zuhair |
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| zuhair... |
| Posted: Thu Nov 05, 2009 2:57 pm |
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Guest
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On Nov 5, 6:47 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
[quote]On Nov 5, 6:28 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
Hi all,
The following is a set theory in FOL with identity and epsilon
membeship that use the concept of hyperregularity.
E: existential quantifier
A: universal quantifier
Define (regular):
x is regular <-> (Ey(yex) ->Ey(yex & ~Ez(zey& zex)))
Define (hyperregular):
x is hyperregular <-> (x is regular & Ay(yex -> y is regular))
Define (subclass):
x subclass y <-> Az ( zex -> zey )
Define (set):
x is a set <-> Ey,z (y is regular & z is hyperregular &
yez & x subclass y)
Axioms:
1. Extensionality: Az (zex<->zey) ->x=y
2. Set existence: Ex ( x is a set )
3.Class Comprehension: If phi is a formula in which at least y is
free, and in which x is not free, then all closures of
Ex Ay ( yex <-> ( y is a set & phi ) )
are axioms.
Define: x={y|phi} <-> Ay ( yex <-> ( y is a set & phi ) )
4. Set Comprehension: if phi is a formula in which c is not free,
having at least y occurring free, and x1,...,xn are its parameters,
and if Q1,...,Qm are all proper subformulas of phi, then
Ac,x1,...,xn
( (x1,...,xn all are sets &
Ay (phi(y) -> y is a set) & c={y|phi} &
Au1,...,um ((u1={y|Q1}&...& um={y|Qm}) -
(~u1=c &...&~um=c)))
-> c is a set ).
are axioms.
Theory definition finished/
Now I claim that this set theory have ZF as a sub-theory of it. We can
have all rules of pairing, union, power, infinity, separation and
replacement of "sets" in this theory.
Zuhair
Though I am not sure, but I think this theory needs axiom of Infinity:
5: Infinity: The class of all finite Von Neumann's ordinals is a set.
Zuhair
[/quote]
I am not sure of Infinity really.
We can use the formula Az((0ez&Am(mez->mU{m}ez)) -> yez)
an substitute it in class comprehension, then it appears that
Infinity
is a theorem of this theory, i.e. no need to axiomatize it, yet I am
not quite sure of this now.
On the other hand if Q is a proper subformula of phi is defined in
such a manner that for Q to be proper subformula of phi then Q should
be a substring of the symbols in phi that is a formula provided that
the remaining
symbols in phi that are not in Q constitute a formula also, then
there
will be no need for axiom 2 (which is ad hoc axiom by the way),
since
y=y will not be a subformula of ~y=y , since what is left of ~y=y
after taking y=y from it is"~" which is not a formula, so according
to
this definition although y=y is a proper substring of ~y=y , yet it
is
not a subformula of ~y=y, because ~ is not a formula. while for
example Let the formula phi be " x is a set & x is ordinal" here "x is
a set" is a proper subformula of phi and also "x is ordinal" is a
proper subformula of phi.
Also one can add some restrictions on the definition of a subformula
like saying that a proper subformula should have the principal
variable of the formula and all parameters of the subformula should
be
a subset of the parameters of the formula, these are also reasonable
restrictions on the definition of subformulas.
Taking all these restrictions on the definition of "proper
subformula", we'll not need axiom 2, and the theory will consist of
only three axioms schemes, and will not possess any ad-hoc-sens in
its
exposition
Zuhair |
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| zuhair... |
| Posted: Thu Nov 05, 2009 2:58 pm |
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Guest
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On Nov 5, 6:47 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
[quote]On Nov 5, 6:28 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
Hi all,
The following is a set theory in FOL with identity and epsilon
membeship that use the concept of hyperregularity.
E: existential quantifier
A: universal quantifier
Define (regular):
x is regular <-> (Ey(yex) ->Ey(yex & ~Ez(zey& zex)))
Define (hyperregular):
x is hyperregular <-> (x is regular & Ay(yex -> y is regular))
Define (subclass):
x subclass y <-> Az ( zex -> zey )
Define (set):
x is a set <-> Ey,z (y is regular & z is hyperregular &
yez & x subclass y)
Axioms:
1. Extensionality: Az (zex<->zey) ->x=y
2. Set existence: Ex ( x is a set )
3.Class Comprehension: If phi is a formula in which at least y is
free, and in which x is not free, then all closures of
Ex Ay ( yex <-> ( y is a set & phi ) )
are axioms.
Define: x={y|phi} <-> Ay ( yex <-> ( y is a set & phi ) )
4. Set Comprehension: if phi is a formula in which c is not free,
having at least y occurring free, and x1,...,xn are its parameters,
and if Q1,...,Qm are all proper subformulas of phi, then
Ac,x1,...,xn
( (x1,...,xn all are sets &
Ay (phi(y) -> y is a set) & c={y|phi} &
Au1,...,um ((u1={y|Q1}&...& um={y|Qm}) -
(~u1=c &...&~um=c)))
-> c is a set ).
are axioms.
Theory definition finished/
Now I claim that this set theory have ZF as a sub-theory of it. We can
have all rules of pairing, union, power, infinity, separation and
replacement of "sets" in this theory.
Zuhair
Though I am not sure, but I think this theory needs axiom of Infinity:
5: Infinity: The class of all finite Von Neumann's ordinals is a set.
Zuhair
[/quote]
I am not sure of Infinity really.
We can use the formula Az((0ez&Am(mez->mU{m}ez)) -> yez)
an substitute it in class comprehension, then it appears that Infinity
is a theorem of this theory, i.e. no need to axiomatize it, yet I am
not quite sure of this now.
On the other hand if Q is a proper subformula of phi is defined in
such a manner
that for Q to be proper subformula of phi then Q should be a substring
of the symbols in phi that is a formula provided that the remaining
symbols in phi that are not in Q constitute a formula also, then there
will be no need for axiom 2 (which is ad hoc axiom by the way), since
y=y will not be a subformula of ~y=y , since what is left of ~y=y
after taking y=y from it is"~" which is not a formula, so according to
this definition although y=y is a proper substring of ~y=y , yet it is
not a subformula of ~y=y, because ~ is not a formula. while for
example Let the formula phi be
" x is a set & x is ordinal" here "x is a set" is a proper subformula
of phi and
also "x is ordinal" is a proper subformula of phi.
Also one can add some restrictions on the definition of a subformula
like saying that a proper subformula should have the principal
variable of the formula and all parameters of the subformula should be
a subset of the parameters of the formula, these are also reasonable
restrictions on the definition of subformulas.
Taking all these restrictions on the definition of "proper
subformula", we'll not need axiom 2, and the theory will consist of
only three axioms schemes, and will not possess any ad-hoc-sens in its
exposition
Zuhair |
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| zuhair... |
| Posted: Thu Nov 05, 2009 8:05 pm |
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Guest
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[quote]
Zuhair
I am not sure of Infinity really.
We can use the formula Az((0ez&Am(mez->mU{m}ez)) -> yez)
an substitute it in class comprehension, then it appears that Infinity
is a theorem of this theory, i.e. no need to axiomatize it, yet I am
not quite sure of this now.
On the other hand if Q is a proper subformula of phi is defined in
such a manner
that for Q to be proper subformula of phi then Q should be a substring
of the symbols in phi that is a formula provided that the remaining
symbols in phi that are not in Q constitute a formula also, then there
will be no need for axiom 2 (which is ad hoc axiom by the way), since
y=y will not be a subformula of ~y=y , since what is left of ~y=y
after taking y=y from it is"~" which is not a formula, so according to
this definition although y=y is a proper substring of ~y=y , yet it is
not a subformula of ~y=y, because ~ is not a formula. while for
example Let the formula phi be
" x is a set & x is ordinal" here "x is a set" is a proper subformula
of phi and
also "x is ordinal" is a proper subformula of phi.
Also one can add some restrictions on the definition of a subformula
like saying that a proper subformula should have the principal
variable of the formula and all parameters of the subformula should be
a subset of the parameters of the formula, these are also reasonable
restrictions on the definition of subformulas.
Taking all these restrictions on the definition of "proper
subformula", we'll not need axiom 2, and the theory will consist of
only three axioms schemes, and will not possess any ad-hoc-sens in its
exposition
Zuhair
[/quote]
MORE IMPORTANT, is that every formula phi that is used in set
comprehension should be of the form "Q & y=y" , so one should specify
that the subformulas Q1,...,Qm mentioned in set comprehension schema
should be subformulas of phi Other than the formula y=y, we can
designate y=y as the trivial formula present in every formula phi used
in set comprehension, so Q1,...,Qm are non trivial formulas of phi.
So for example we can use the formula y=y & y=y
but we cannot use the formula x is a set & y=y because it would be a
redundant formula, any formula which is true of all sets have a
subformula in it other than y=y
cannot be used, since it would be redundant.
Zuhair |
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| zuhair... |
| Posted: Fri Nov 06, 2009 2:13 pm |
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Guest
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On Nov 6, 1:05 am, zuhair <zaljo... at (no spam) gmail.com> wrote:
[quote]Zuhair
I am not sure of Infinity really.
We can use the formula Az((0ez&Am(mez->mU{m}ez)) -> yez)
an substitute it in class comprehension, then it appears that Infinity
is a theorem of this theory, i.e. no need to axiomatize it, yet I am
not quite sure of this now.
On the other hand if Q is a proper subformula of phi is defined in
such a manner
that for Q to be proper subformula of phi then Q should be a substring
of the symbols in phi that is a formula provided that the remaining
symbols in phi that are not in Q constitute a formula also, then there
will be no need for axiom 2 (which is ad hoc axiom by the way), since
y=y will not be a subformula of ~y=y , since what is left of ~y=y
after taking y=y from it is"~" which is not a formula, so according to
this definition although y=y is a proper substring of ~y=y , yet it is
not a subformula of ~y=y, because ~ is not a formula. while for
example Let the formula phi be
" x is a set & x is ordinal" here "x is a set" is a proper subformula
of phi and
also "x is ordinal" is a proper subformula of phi.
Also one can add some restrictions on the definition of a subformula
like saying that a proper subformula should have the principal
variable of the formula and all parameters of the subformula should be
a subset of the parameters of the formula, these are also reasonable
restrictions on the definition of subformulas.
Taking all these restrictions on the definition of "proper
subformula", we'll not need axiom 2, and the theory will consist of
only three axioms schemes, and will not possess any ad-hoc-sens in its
exposition
Zuhair
MORE IMPORTANT, is that every formula phi that is used in set
comprehension should be of the form "Q & y=y" , so one should specify
that the subformulas Q1,...,Qm mentioned in set comprehension schema
should be subformulas of phi Other than the formula y=y, we can
designate y=y as the trivial formula present in every formula phi used
in set comprehension, so Q1,...,Qm are non trivial formulas of phi.
So for example we can use the formula y=y & y=y
but we cannot use the formula x is a set & y=y because it would be a
redundant formula, any formula which is true of all sets have a
subformula in it other than y=y
cannot be used, since it would be redundant.
Zuhair
[/quote]
Sorry that was messed up.
Set comprehensions schema should be written in the following manner:
4. Set Comprehension: if phi is a formula in which c is not free,
having at least y occurring free, and x1,...,xn are its parameters,
and if Q1,...,Qm are all proper subformulas of phi, then
Ac,x1,...,xn
( (x1,...,xn all are sets &
Ay (phi(y) -> y is a set) & c={y|phi} &
Au1,...,um ((u1={y|Q1}&...& um={y|Qm}) ->
(~u1=c &...&~um=c)) & ~c={y|y=y})
-> c is a set ).
are axioms.
of course in this way we will need the ad hoc axiom 2.
Zuhair |
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