Re: Why Regularity?


Zuhair wrote:
I have a question.

Read the aximatic system above.

Now if I make anther set theory keeping the following axioms as axioms:

Extentiality,Irregularity,Uniqueness,Unrestricted
comprehension,Infinity.

Can the other axioms be theorums in this theory.

It seems to me that all the other axioms:
Completion,Pairing,Union,Separation,Replacement,Power,Choice, and
perhaps even Infinity?

can be derived using the axiom of unrestriced comprehsion ( see this
axiom above, it is not the same as the well known axiom of unrestriced
comprehsion, it contains a second order logic statment that quantify
over P(y), it states that there should be at least one y for which P
holds, or put in other equivalent words, P(y) should hold for some y.
This condition together with the other axioms will prevent Russell
paradox in this set theory.

Is the above correct?

Zuhair

As a continuation to this post, if it can be proved that from the axiom
of unrestricted comprehension all the other axioms I have mentioned
will be just theorums , then I think I have reached into the set theory
that I always wanted, it is simply the following:


- Irregular Set Theory-

Primitive e

1) Axiom of Extentiality: Ax,Ay: x=y <-> (Az:zex <-> zey)
2) Axiom of Irregularity: Ax (xex)
3) Axiom of Uniqueness: EQ,Ax (Qex)
4) Axiom of Comprehension: Ex,Ay:y=/=Q, yex<->P(y)

From this set theory, some of what is regarded as axioms in ZFC, with
slight modefication would be theorums here.

Theorum 1. ExAy (yex)
Theorum 2) ~ExAy ~(yex)
Theorum 3) Pairing : as in ZFC
Theorum 4) Union: as in ZFC
Theorum 5) Separation: AyExAz:z=/=Q ( zex <-> zey /\ P(y) ).
Theorum 6) Replacement: AxE!y:P(x,y) -> AaEbAy:y=/=Q,yeb<->Exea:P(x,y)
Theorum 7) Power: as in ZFC
Theorum 8) Choice: as in ZFC
Theorum 9) Infinity: EN:QeN /\(Ax:xeN -> y:(Am:mex->mey)/\(x'ey)) eN )
x' is the complementary set of x , i.e. x':Az:z=/=Q (zex' <-> z!ex)
N= the set of all ordinals

According to 9) the set of all ordinals exist. I think the cardinal of
all cardinals would also be
a theorum in this set theory.

In this set theory it can be proved that for some sets, they are
identical as their power sets.
Also Cantor's proof of non bijectability of x to P(x) is not valid
here.

Zuhair

.

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