THIS IS ALL Z.F.C. IS ....... Y = { X | X e Z & P(X,Y) }

[Graham Cooper]

1. Naive Set Theory

y = { x | P(x) }

_______________________________

2. Zermelo Frankel Set Theory (ZFC)

ZFC Axiom Of Specification
A(z): A(p1,p2..pn):
E(y): A(x): x e y <-> (x e z & P(x,y,p1,p2..pn))

MORE SIMPLY
y = { x | x e z & P(x,y) }

_______________________________

3. Cooper Set Theory

E(y) y = { x | P(x,y) } <-> !PROOF( !E(y) y = { x | P(x,y) } )

PROOF(C) <-> C v (PROOF(A) ^ PROOF(B) ^ (A^B->C))

_______________________________

Mr Cooper

[Tonico]

Idiot

[Graham Cooper]

On Apr 16, 10:39 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> 1. Naive Set Theory
>
> y = { x | P(x) }
>
> _______________________________
>
> 2. Zermelo Frankel Set Theory  (ZFC)
>
> ZFC Axiom Of Specification
> A(z): A(p1,p2..pn):
> E(y): A(x): x e y <-> (x e z & P(x,y,p1,p2..pn))
>
> MORE SIMPLY
> y = { x | x e z & P(x,y) }
>

If we let NAIVE SET THEORY set instances be defined via some predicate
P

E(Y) X e Y <-> P(X,Y)

THEN ZFC is merely the weaker formula

E(Y) X e Y -> P(X,Y)

********************

We us an Implication for ZFC
instead of a BiConditional for Naive S.T.

That's the only difference!

PROOF:

GIVEN:
Y C Z
<->
X e Y -> X e Z

GIVEN:
E(Y) X e Y -> P(X,Y)

THEN:
X e Y -> (X e Z) & P(X,Y)

GIVEN:
A(x,y)
y = { x | x e Y }

THEN

y = { x | x e z & P(x,y) }

QED


ZFC is derived from one side of the implication of NAIVE SET THEORY

****************

NAIVE SET THEORY
E(Y) X e Y <-> P(X,Y)

ZFC
E(Y) X e Y -> P(X,Y)

Basically a formula to define SUBSETS! Pairing and what not were
tacked on to compensate for this weakness and the whole "constructable
Universe" was invented thereafter!

Herc

[Don Stockbauer]

Stockbauer's Set Theory - let's throw boomerangs instead!!!!