An Axiomatic system Equivalent to ZFC.
Zalj...@gmail.com
Though I am not sure weather the following axiomatic system is
equivalent to that of ZFC, but I would present it in order to be
discussed by anybody who is interested in this subject.
1)Axiom of Regularity: as in Z
2)Axiom of Extensionality: as in Z
Definition: x is transitive <-> Anm(nem&mex->nex)
3)Axiom of Transitive closure:
AxE!y( Am(mex->mey) & y is transitive &
Am( (mey & ~mex) -> Ez( zey & mez ) ) ).
Definition:
y=Tc(x) <-> ( Am(mex->mey) & y is transitive &
Am( (mey & ~mex) -> Ez( zey & mez ) ) ).
Definition:
x is P_defined <-> Ay(yex<->P(y))
were P is a formula in which x is not free.
4)Axiom schema of Comprehension:
Ex ( x is P_defined <-> Ay( yeTc(x) -> ~ y is P_defined ) )
is an axiom.
5)Axiom of Choice: as in ZC.
I think that this system is equivalent to ZFC. Though it depends in
it's essence on Regularity.
It is clear that pairing , power are theorems, since they satisfy
comprehension, and for the same reason Separation is a theorem schema
in this theory, from this we can easily prove union( from separation
and Transitive closure ) .
Infinity is proved because the set of all hereditary finite sets is
proved from Comprehension and it is easily proved to be infinite.
Omega is proved thereafter from separation on the set of all
hereditary finite sets using the formula F(y)<->y is ordinal.
IF ( and this is a big IF ) this axiomatic system proves to be
equivalent to ZFC, then it should replace the classical one, since it
is more elegant and provides an more systematic approach than the ad
hoc axioms of ZFC (Extensionality,Replacement Power,Union,Infinity,
+/- Regularity and choice ), and it would be even better and simpler
than the alternative axiomatization of ZFC that Ackermann's set theory
provide.
Zuhair
Zalj...@gmail.com
Sorry for this post.
Axiom 4 cannot work.
The only system of alternative axiomatization to ZFC that I have is a
superficial one that bears no importance and it is the following:
Axiom of Regularity: as in Z
Axiom of Extensionality: as in Z
Axiom of Replacement(strong version): as in ZF
Axiom of Power: as in Z
Axiom of Transitive closure:
AxE!y( Am(mex->mey) & y is transitive &
Am( (mey & ~mex) -> Ez( zey & mez ) ) ).
Definition:
y=Tc(x) <-> ( Am(mex->mey) & y is transitive &
Am( (mey & ~mex) -> Ez( zey & mez ) ) ).
Definition:
x is hereditary finite <-> ( x is finite & Ay ( yeTc(x) -> y is
finite ) ).
Axiom of Infinity: ExAy ( yex <-> y is hereditary finite ).
Axiom of Choice: As in ZC
Pairing,Separation,Empty are all theorems in this theory.
Union comes from Separation and transitive closure.
Infinity can be easily proved from axiom of infinity and regularity.
Zuhair
Zalj...@gmail.com
>
> Though I am not sure weather the following axiomatic system is
> equivalent to that of ZFC, but I would present it in order to be
> discussed by anybody who is interested in this subject.
>
> 1)Axiom of Regularity: as in Z
>
> 2)Axiom of Extensionality: as in Z
>
> Definition: x is transitive <-> Anm(nem&mex->nex)
>
> 3)Axiom of Transitive closure:
>
> AxE!y( Am(mex->mey) & y is transitive &
> Am( (mey & ~mex) -> Ez( zey & mez ) ) ).
>
> Definition:
>
> y=Tc(x) <-> ( Am(mex->mey) & y is transitive &
> Am( (mey & ~mex) -> Ez( zey & mez ) ) ).
>
> Definition:
>
> x is P_defined <-> Ay(yex<->P(y))
>
> were P is a formula in which x is not free.
>
> 4)Axiom schema of Comprehension:
>
> Ex ( x is P_defined <-> Ay( yeTc(x) -> ~ y is P_defined ) )
>
> is an axiom.
This cannot work.
perhaps what should be done is to add another primitive to the
language of this theory that is V. and this would stand for the class
of all sets.
Then we restrict Regularity to members of V only.
and modify 4 to
Ex ( (x is P_defined & xeV) <-> ( Ay( yeTc(x) -> ~ y=x ) & ~P(x) ) )
is an axiom.
Perhaps this might work.
>
> 5)Axiom of Choice: as in ZC.
This should be limited within V.