Re: ZFC in 4 Axioms.


zuhair wrote:
MoeBlee wrote:
zuhair wrote:

Definition:

Ax(x is P_embeded <-> (Ay: y e x <-> P(y)) /\ P(x))

What does 'P' range over? Formulas? If so, then I'm concerned whether
we can justify your definition even as a definitional schema. But let's
see what we find.

But first, please use parentheses rather than the ':' colon symbol. It
will be much clearer.

Now, I don't know whether you want:

x is P_embedded
stands for
Ay(yex <-> (P(y) & P(x)))

or

x is P_embedded
stands for
Ay(yex <-> P(y)) & P(x)

or

x is P_embeddd
stands for
Ay yex <-> (P(y) & P(x))

or

x is P_embdded
stands for
(Ay yex <-> P(y)) & P(x)

MoeBlee

let me present it in an informal manner and you choose.

x is embeded with respect to formula P means that every member of x
satisfy P and x itself satisfy P, and every z : z!ex /\ z!=x then
P(z)is false.

Let me try write it in symboles:

x is P_embeded <-> ( [ Ay(yex->P(y)) ] /\ [P(x)] /\ [ Az((z!ex /\ z!=x)
-> ~P(z)) ] )

I think this is the best way to put it in symboles you people here
odour and never work without them.

Zuhair

Now the axiomatic system is.

-Small Set Theory-

Primitive e

Definition:

x is P_embeded<->([Ay(yex->P(y))]/\[P(x)]/\[Az((z!ex /\ z!=x)->~P(z))])

Axioms

Ax.1: Extensionality: As in ZFC
Ax.2: Comprehension:Ex Ay ( yex<-> (P(y) /\ ~(x is P_embeded)) )
Ax.3: Exclusion: Ax Ay ( yex<-> (P(y) /\ ~(x is P_embeded)) )
Ax.4: Infinity:As in ZFC.

Theorums:

1) Universe. E!v Ay (yev).

Definition: V={y|y=y}, so V is the set of all other sets.
and it is the unvierse of all sets in this theory.


2) Pairing:Aa Ab Ex Ay ( yex<->( (y=a v y=b) /\ ~(y is (y=a v
y=b)_embeded) )
)
3) Union:Ex Ay ( yex<-> ((Eu:yeu /\ uex) /\ ~(y is (Eu:yeu /\
uex)_embeded) )
4) Separation: Ax Ey Az ( zey<-> ((zex/\P(z)) /\ ~(z is
(zex/\P(z))_embeded))
)
5) Replacement:(AxE!y:P(x,y))->AaEbAy:yeb<->( (Exea:P(x,y) /\
~(y is (Exea:P(x,y)_embeded)).
6) Power:Ax EP(x) Az ( zeP(x)<->( (Ay:yez->yex) /\ ~(z is
(Ay:yez->yex)_
embeded)) )


Zuhair

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