Theory /\ is the set of all sentences entailed by (using FOL with identity and the primitives "in" and "V" were V is a constant) the following non logical axioms:
Definition: x is a set iff x in V Definition: x is a class iff [ x in V or ~ x in V ]
So we have: for all x : x is a class.
1) Axiom of Extensionality:
forall z (z in A <-> z in B) -> A=B
2) Axiom of complementary class.
forall A, Exists X : forall y ( y in X <-> ~ y in A )
Define: X=A' iff forall y ( y in X <-> ~ y in A )
A' is read as 'the complementary class of A'
3) Axiom of Intersection:
forall A, Exists X : forall y ( y in X <-> forall z ( z in A -> y in z ) )
Define: X= /\ A iff forall y ( y in X <-> forall z ( z in A -> y in z ) )
/\ A is read as 'the intersectional class of A'.
4) All axioms of Z except regularity and sepration.
Define: X = { A,B } iff forall y ( y in X <-> (y=A or y=B) )
{A,B} is read as the unordered pair of A and B.
Define: X= Union A iff forall y ( y in X <-> Exists z ( z in A and y in z ) ).
Union A is read as "the class union of A".
Define: X= Power A iff forall y ( y in X <-> forall z ( z in y -> z in A ) ).
Power A is read as "the power class of A".
5) Axiom of set existence: Exists X : X in V
6) Axiom of reflexiseness of V: forall X in V , forall y ( y in X -> y in V ). i.e: any member of a set is a set. or: any set is a subclass of V.
7.1) Axiom of pairing of sets: The unordered pair of two sets is a set. 7.2) Axiom of set union: The class union of a set is a set. 7.3) Axiom of power set: The power class of a set is a set. 7.4) Axiom of Infinity:
Exists N in V ( 0 in N and forall y (y in N -> yU{y} in N) ).
7.5) Axiom schema of separation for sets: If P is a formula in which X is not free, then all closures of
forall A in V, Exists X in V, forall y ( y in X <-> ( y in A and P(y) ) )
are axioms.
7.6) Axiom schema of replacement for sets: If P is a formula in which B is not free, then all closures of
[(forall x in V, Exists! y in V:P(x,y)) and (forall x in V ( P(x,y) -> y in V ))]
-> forall A in V ,Exists B in V, forall y ( y in B <-> Exists x in A: P(x,y) ).
are axioms.
7.7) Axiom of Regularity for sets: For every non empty set there should exist an element that is disjoint of it.
7.8) Axiom of choice for sets: V is well orderable.
8) Axiom schema of class comprehension over V: if P is a formula in which X is not free, then all closure of
Exists X: forall y ( y in X <-> ( y in V and P(y) ) ).
are axioms.
9) Axiom schema of partial replacement for classes: If P is a formula in which B is not free, then all closures of:
[forall x in V, Exists! y : P(x,y)] -> forall A subset_of V, Exists B, forall y ( y in B <-> Exists x in A:P(x,y) ).
are axioms.
/Theory definition finished
Perhaps, and I say perhaps Axiom 8 is redundant.
I think that we can of course reduce the number of axioms in this theory, however I think the above presentation is a good one to begin with.
This theory do not have Ur-elements classes in this theory are divided into: sets: which are classes that are members of V. proper classes: which are subsets of V but not members of V. ultraclasses: which are not subsets of V, but smaller of equal in size to V. universal classes: like U the class of all classes, and classes that are near to U in size, like U except 0 for example,etc...
Now the question here is: Is this theory consistent?
> Theory /\ is the set of all sentences entailed by (using FOL with > identity and the primitives "in" and "V" were V is a constant) the > following non logical axioms:
> Definition: x is a set iff x in V > Definition: x is a class iff [ x in V or ~ x in V ]
> So we have: for all x : x is a class.
> 1) Axiom of Extensionality:
> forall z (z in A <-> z in B) -> A=B
> 2) Axiom of complementary class.
> forall A, Exists X : forall y ( y in X <-> ~ y in A )
> Define: X=A' iff forall y ( y in X <-> ~ y in A )
> A' is read as 'the complementary class of A'
> 3) Axiom of Intersection:
> forall A, Exists X : forall y ( y in X <-> forall z ( z in A -> y in > z ) )
> Define: X= /\ A iff forall y ( y in X <-> forall z ( z in A -> y in > z ) )
> /\ A is read as 'the intersectional class of A'.
> 4) All axioms of Z except regularity and sepration.
> Define: X = { A,B } iff forall y ( y in X <-> (y=A or y=B) )
> {A,B} is read as the unordered pair of A and B.
> Define: X= Union A iff forall y ( y in X <-> Exists z ( z in A and > y > in z ) ).
> Union A is read as "the class union of A".
> Define: X= Power A iff forall y ( y in X <-> forall z ( z in y -> z > in A ) ).
> Power A is read as "the power class of A".
> 5) Axiom of set existence: Exists X : X in V
> 6) Axiom of reflexiseness of V: forall X in V , forall y ( y in X -> > y > in V ). > i.e: any member of a set is a set. > or: any set is a subclass of V.
> 7.1) Axiom of pairing of sets: The unordered pair of two sets is a > set. > 7.2) Axiom of set union: The class union of a set is a set. > 7.3) Axiom of power set: The power class of a set is a set. > 7.4) Axiom of Infinity:
> Exists N in V ( 0 in N and forall y (y in N -> yU{y} in N) ).
> 7.5) Axiom schema of separation for sets: If P is a formula in which X > is not free, then all closures of
> forall A in V, Exists X in V, forall y ( y in X <-> ( y in A and > P(y) ) )
> are axioms.
> 7.6) Axiom schema of replacement for sets: If P is a formula in which > B is not free, then all closures of
> [(forall x in V, Exists! y in V:P(x,y)) > and (forall x in V ( P(x,y) -> y in V ))]
> -> forall A in V ,Exists B in V, forall y ( y in B <-> Exists x in A: > P(x,y) ).
> are axioms.
> 7.7) Axiom of Regularity for sets: For every non empty set there > should exist an element that is disjoint of it.
> 7.8) Axiom of choice for sets: V is well orderable.
> 8) Axiom schema of class comprehension over V: if P is a formula in > which X is not free, then all closure of
> Exists X: forall y ( y in X <-> ( y in V and P(y) ) ).
> are axioms.
> 9) Axiom schema of partial replacement for classes: If P is a formula > in which B is not free, then all closures of:
> [forall x in V, Exists! y : P(x,y)] -> > forall A subset_of V, Exists B, forall y ( y in B <-> Exists x in > A:P(x,y) ).
> are axioms.
> /Theory definition finished
> Perhaps, and I say perhaps Axiom 8 is redundant.
> I think that we can of course reduce the number of axioms in this > theory, however I think the above presentation is a good one to begin > with.
> This theory do not have Ur-elements > classes in this theory are divided into: > sets: which are classes that are members of V. > proper classes: which are subsets of V but not members of V. > ultraclasses: which are not subsets of V, but smaller of equal in size > to V. > universal classes: like U the class of all classes, and classes that > are near to U in size, like U except 0 for example,etc...
> Now the question here is: Is this theory consistent?
> Zuhair
Yes, I believe that I can prove that this theory is consistent provided it is consistent with ZFC that an inaccessible cardinal exists and I may post a proof of this later.
> Theory /\ is the set of all sentences entailed by (using FOL with > identity and the primitives "in" and "V" were V is a constant) the > following non logical axioms:
> Definition: x is a set iff x in V > Definition: x is a class iff [ x in V or ~ x in V ]
> So we have: for all x : x is a class.
> 1) Axiom of Extensionality:
> forall z (z in A <-> z in B) -> A=B
> 2) Axiom of complementary class.
> forall A, Exists X : forall y ( y in X <-> ~ y in A )
> Define: X=A' iff forall y ( y in X <-> ~ y in A )
> A' is read as 'the complementary class of A'
> 3) Axiom of Intersection:
> forall A, Exists X : forall y ( y in X <-> forall z ( z in A -> y in > z ) )
> Define: X= /\ A iff forall y ( y in X <-> forall z ( z in A -> y in > z ) )
> /\ A is read as 'the intersectional class of A'.
> 4) All axioms of Z except regularity and sepration.
> Define: X = { A,B } iff forall y ( y in X <-> (y=A or y=B) )
> {A,B} is read as the unordered pair of A and B.
> Define: X= Union A iff forall y ( y in X <-> Exists z ( z in A and > y > in z ) ).
> Union A is read as "the class union of A".
> Define: X= Power A iff forall y ( y in X <-> forall z ( z in y -> z > in A ) ).
> Power A is read as "the power class of A".
> 5) Axiom of set existence: Exists X : X in V
> 6) Axiom of reflexiseness of V: forall X in V , forall y ( y in X -> > y > in V ). > i.e: any member of a set is a set. > or: any set is a subclass of V.
> 7.1) Axiom of pairing of sets: The unordered pair of two sets is a > set. > 7.2) Axiom of set union: The class union of a set is a set. > 7.3) Axiom of power set: The power class of a set is a set. > 7.4) Axiom of Infinity:
> Exists N in V ( 0 in N and forall y (y in N -> yU{y} in N) ).
> 7.5) Axiom schema of separation for sets: If P is a formula in which X > is not free, then all closures of
> forall A in V, Exists X in V, forall y ( y in X <-> ( y in A and > P(y) ) )
> are axioms.
> 7.6) Axiom schema of replacement for sets: If P is a formula in which > B is not free, then all closures of
> [(forall x in V, Exists! y in V:P(x,y)) > and (forall x in V ( P(x,y) -> y in V ))]
> -> forall A in V ,Exists B in V, forall y ( y in B <-> Exists x in A: > P(x,y) ).
> are axioms.
> 7.7) Axiom of Regularity for sets: For every non empty set there > should exist an element that is disjoint of it.
> 7.8) Axiom of choice for sets: V is well orderable.
> 8) Axiom schema of class comprehension over V: if P is a formula in > which X is not free, then all closure of
> Exists X: forall y ( y in X <-> ( y in V and P(y) ) ).
> are axioms.
> 9) Axiom schema of partial replacement for classes: If P is a formula > in which B is not free, then all closures of:
> [forall x in V, Exists! y : P(x,y)] -> > forall A subset_of V, Exists B, forall y ( y in B <-> Exists x in > A:P(x,y) ).
> are axioms.
Note: Subset in the above axiom schema means subclass.
> Perhaps, and I say perhaps Axiom 8 is redundant.
> I think that we can of course reduce the number of axioms in this > theory, however I think the above presentation is a good one to begin > with.
> This theory do not have Ur-elements > classes in this theory are divided into: > sets: which are classes that are members of V. > proper classes: which are subsets of V but not members of V. > ultraclasses: which are not subsets of V, but smaller of equal in size > to V. > universal classes: like U the class of all classes, and classes that > are near to U in size, like U except 0 for example,etc...
> Now the question here is: Is this theory consistent?
> Zuhair
I am thinking of adding an axiom that is similar to inverse intersection. An axiom that reveal the maximal class the intersection of which is C for example.
Axiom of maximal inverse intersection.
forall C, Exists X : forall y ( y in X <-> C subclass of y ).
> > Theory /\ is the set of all sentences entailed by (using FOL with > > identity and the primitives "in" and "V" were V is a constant) the > > following non logical axioms:
> > Definition: x is a set iff x in V > > Definition: x is a class iff [ x in V or ~ x in V ]
> > So we have: for all x : x is a class.
> > 1) Axiom of Extensionality:
> > forall z (z in A <-> z in B) -> A=B
> > 2) Axiom of complementary class.
> > forall A, Exists X : forall y ( y in X <-> ~ y in A )
> > Define: X=A' iff forall y ( y in X <-> ~ y in A )
> > A' is read as 'the complementary class of A'
> > 3) Axiom of Intersection:
> > forall A, Exists X : forall y ( y in X <-> forall z ( z in A -> y in > > z ) )
> > Define: X= /\ A iff forall y ( y in X <-> forall z ( z in A -> y in > > z ) )
> > /\ A is read as 'the intersectional class of A'.
> > 4) All axioms of Z except regularity and sepration.
> > Define: X = { A,B } iff forall y ( y in X <-> (y=A or y=B) )
> > {A,B} is read as the unordered pair of A and B.
> > Define: X= Union A iff forall y ( y in X <-> Exists z ( z in A and > > y > > in z ) ).
> > Union A is read as "the class union of A".
> > Define: X= Power A iff forall y ( y in X <-> forall z ( z in y -> z > > in A ) ).
> > Power A is read as "the power class of A".
> > 5) Axiom of set existence: Exists X : X in V
> > 6) Axiom of reflexiseness of V: forall X in V , forall y ( y in X -> > > y > > in V ). > > i.e: any member of a set is a set. > > or: any set is a subclass of V.
> > 7.1) Axiom of pairing of sets: The unordered pair of two sets is a > > set. > > 7.2) Axiom of set union: The class union of a set is a set. > > 7.3) Axiom of power set: The power class of a set is a set. > > 7.4) Axiom of Infinity:
> > Exists N in V ( 0 in N and forall y (y in N -> yU{y} in N) ).
> > 7.5) Axiom schema of separation for sets: If P is a formula in which X > > is not free, then all closures of
> > forall A in V, Exists X in V, forall y ( y in X <-> ( y in A and > > P(y) ) )
> > are axioms.
> > 7.6) Axiom schema of replacement for sets: If P is a formula in which > > B is not free, then all closures of
> > [(forall x in V, Exists! y in V:P(x,y)) > > and (forall x in V ( P(x,y) -> y in V ))]
> > -> forall A in V ,Exists B in V, forall y ( y in B <-> Exists x in A: > > P(x,y) ).
> > are axioms.
> > 7.7) Axiom of Regularity for sets: For every non empty set there > > should exist an element that is disjoint of it.
> > 7.8) Axiom of choice for sets: V is well orderable.
> > 8) Axiom schema of class comprehension over V: if P is a formula in > > which X is not free, then all closure of
> > Exists X: forall y ( y in X <-> ( y in V and P(y) ) ).
> > are axioms.
> > 9) Axiom schema of partial replacement for classes: If P is a formula > > in which B is not free, then all closures of:
> > [forall x in V, Exists! y : P(x,y)] -> > > forall A subset_of V, Exists B, forall y ( y in B <-> Exists x in > > A:P(x,y) ).
> > are axioms.
> Note: Subset in the above axiom schema means subclass.
> > /Theory definition finished
> > Perhaps, and I say perhaps Axiom 8 is redundant.
> > I think that we can of course reduce the number of axioms in this > > theory, however I think the above presentation is a good one to begin > > with.
> > This theory do not have Ur-elements > > classes in this theory are divided into: > > sets: which are classes that are members of V. > > proper classes: which are subsets of V but not members of V. > > ultraclasses: which are not subsets of V, but smaller of equal in size > > to V. > > universal classes: like U the class of all classes, and classes that > > are near to U in size, like U except 0 for example,etc...
> > Now the question here is: Is this theory consistent?
> > Zuhair
> I am thinking of adding an axiom that is similar to inverse > intersection. An axiom that reveal the maximal class the intersection > of which is C for example.
> Axiom of maximal inverse intersection.
> forall C, Exists X : forall y ( y in X <-> C subclass of y ).
> Zuhair- Hide quoted text -
> - Show quoted text -- Hide quoted text -
> - Show quoted text -
Also I will add axiom schema of equivalence classes:
If R is equivlance relation, then all closures of
forall A, Exists X, forall y ( y in X <-> y R A )
is an axiom.
equivalence relation mean a relation that is reflexive,symmetric and transitive.
By this schema, on can have the equivalence class of all classes bijective to A. also one can have equivalence class of all classes ordinally isomorphic to A.
In this way one can define cardinal numbers as these equivalence classes of bijective classes, and also define ordinal numbers as equivalance classes of ordinally isomorphic classes. i.e. Frege's method can be adopted here in this thoery.
This theory seem's to be nearer actually to Cantor's set theory, since Cantor's original theory had a class of all classes in it (see Cantor's paradox), and also Cantor defined cardinality and ordinality in terms of equivalence classes.
So this theory is the nearest to Cantor's concepts of set theory, not ZFC.
> Also I will add axiom schema of equivalence classes:
> If R is equivlance relation, then all closures of
> forall A, Exists X, forall y ( y in X <-> y R A )
> is an axiom.
I don't think that works. You're in the meta-language talking about R being an equivalence relation, which actually belongs in the object- language. I don't see that it can't be a direct axiom rather than a schema. So maybe this instead:
R is an equivalence relation -> AzExAy(yex <-> <y z>eR)
> > Also I will add axiom schema of equivalence classes:
> > If R is equivlance relation, then all closures of
> > forall A, Exists X, forall y ( y in X <-> y R A )
> > is an axiom.
> I don't think that works. You're in the meta-language talking about R > being an equivalence relation, which actually belongs in the object- > language. I don't see that it can't be a direct axiom rather than a > schema. So maybe this instead:
> R is an equivalence relation -> AzExAy(yex <-> <y z>eR)
> MoeBlee
a relation is not necessarily an object. so I think the way I put it is the right one.
> > > Also I will add axiom schema of equivalence classes:
> > > If R is equivlance relation, then all closures of
> > > forall A, Exists X, forall y ( y in X <-> y R A )
> > > is an axiom.
> > I don't think that works. You're in the meta-language talking about R > > being an equivalence relation, which actually belongs in the object- > > language. I don't see that it can't be a direct axiom rather than a > > schema. So maybe this instead:
> > R is an equivalence relation -> AzExAy(yex <-> <y z>eR) > a relation is not necessarily an object. so I think the way I put it > is the right one.
It doesn't parse correctly. You use 'R' as a variable of the object language, but mention R ITSELF in the meta-language. That doesn't make sense, as well as there is no need for that kind of thing.
And what do you think is lacking in this much simpler formulation, just a plain axiom:
R is an equivalence relation -> AzExAy(yex <-> <y z>eR)
> > On Jul 25, 3:36 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> > > On Jul 25, 3:04 pm, Zaljo...@gmail.com wrote:
> > > > Also I will add axiom schema of equivalence classes:
> > > > If R is equivlance relation, then all closures of
> > > > forall A, Exists X, forall y ( y in X <-> y R A )
> > > > is an axiom.
> > > I don't think that works. You're in the meta-language talking about R > > > being an equivalence relation, which actually belongs in the object- > > > language. I don't see that it can't be a direct axiom rather than a > > > schema. So maybe this instead:
> > > R is an equivalence relation -> AzExAy(yex <-> <y z>eR) > > a relation is not necessarily an object. so I think the way I put it > > is the right one.
> It doesn't parse correctly. You use 'R' as a variable of the object > language, but mention R ITSELF in the meta-language. That doesn't make > sense, as well as there is no need for that kind of thing.
> And what do you think is lacking in this much simpler formulation, > just a plain axiom:
> R is an equivalence relation -> AzExAy(yex <-> <y z>eR)
> > On Jul 25, 3:36 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> > > On Jul 25, 3:04 pm, Zaljo...@gmail.com wrote:
> > > > Also I will add axiom schema of equivalence classes:
> > > > If R is equivlance relation, then all closures of
> > > > forall A, Exists X, forall y ( y in X <-> y R A )
> > > > is an axiom.
> > > I don't think that works. You're in the meta-language talking about R > > > being an equivalence relation, which actually belongs in the object- > > > language. I don't see that it can't be a direct axiom rather than a > > > schema. So maybe this instead:
> > > R is an equivalence relation -> AzExAy(yex <-> <y z>eR) > > a relation is not necessarily an object. so I think the way I put it > > is the right one.
> It doesn't parse correctly. You use 'R' as a variable of the object > language, but mention R ITSELF in the meta-language. That doesn't make > sense, as well as there is no need for that kind of thing.
> And what do you think is lacking in this much simpler formulation, > just a plain axiom:
> R is an equivalence relation -> AzExAy(yex <-> <y z>eR)
What is lacking is that R here is an object. While I mean relations that are not necessarily objects. And in FOL we cannot quantify over such relations.
