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Topic: -[ PROVABLE SET THEORY ]-
Replies: 0

 Graham Cooper Posts: 4,495 Registered: 5/20/10
-[ PROVABLE SET THEORY ]-
Posted: Mar 4, 2013 6:58 PM

PROVABLE SET THEORY has 1 axiom:

a Self-Consistent Tautology based on Naive Set Theory.

----------------P.S.T.------------------
E(S) A(X) [XeS <-> P(X)]
<->
~(~E(S) A(X) [XeS <-> P(X)] )
------------------------------------------

The P.S.T. Axiom is the Tautology!
formula <-> ~~formula

-------------------------------------------

Based on NAIVE SET THEORY AXIOM
E(S) A(X) [XeS <-> P(X)]

Since NOT(EXIST(RUSSELL'S SET))
where
P(X) <-> X ~e X

EXIST(RUSSELL'S SET) is barred.

------------------------------------------

The is no Universe V!
The Transitive Closure of P.S.T. is {}
All Sets Existence must be assumed!
N.S.T and Z.F.C AXIOM of SPECIFICATION
assert Existence of Sets for certain predicates!
E(S) .....

----------------------------------------

P.S.T. is just a bi-conditional, a RESTRICTION that contradicts

THE ASSUMPTION OF RUSSELL'S SET!

i.e. Only Sets that are PROVEN OR ASSUMED TO EXIST EXIST!

(no Set can be proven to exist)

Not only does RUSSELL'S SET DISAPPEAR, so does EVERY SET!

The difference is other sets can be assumed, Russell's Set cannot be
assumed.

--------------------------------------

A Set Exists iff there is no proof otherwise!

E(S) A(X) [XeS <-> P(X)]
IFF
~(~E(S) A(X) [XeS <-> P(X)] )

--------------------------------------

E(S) A(X) [XeS <-> P(X)]
<->
~(~E(S) A(X) [XeS <-> P(X)] )

-->

E(RS) A(X) [XeRS <-> P(X)]
<->
~(~E(RS) A(X) [XeRS <-> P(X)] )

-->

E(RS) A(X) [XeRS <-> X~eX ]
<->
~(~E(RS) A(X) [XeRS <-> X~eX ])

-->

E(RS) A(X) [XeRS <-> X~eX ]
<->
~(~E(RS) A(X) [RSeRS <-> RS~eRS] )

-->

E(RS) A(X) [XeRS <-> X~eX ]
<->
~(~E(RS) A(X) [f<->~f] )

-->

E(RS) A(X) [XeRS <-> X~eX ]
<->
~(~E(RS) A(X) FALSE )

-->

E(RS) A(X) [XeRS <-> X~eX ]
<->
~TRUE

-->

~E(RS) A(X) [XeRS <-> X~eX ]

FROM A TAUTOLOGY EMERGES A TRUE FACT!

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

ASSUMING
E(N) A(X) [XeN <-> P(X,N)]

WHERE
P(x, N) <-> (x=0) v (E(z) x=S(z) & P(z, N))

replaces the AXIOM OF INFINITY

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

E(Y) Y={x|P(x)} <-> PROVABLE( E(Y) Y={x|P(x)} )
NOT(PROVABLE(T)) <-> DERIVE(NOT(T))

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

G. COOPER (BINFTECH)

-[ PROVABLE SET THEORY ]-

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