Ackermann like set theory.
- From: zuhair <zaljohar@xxxxxxxxx>
- Date: Thu, 07 Jun 2007 19:53:26 -0700
The following theory T looks like Ackermann's but it is not.
The language: LT
1st order logic with identity and constant V.
The axioms
The axioms of this theory set theory, collectively referred to as T,
consists of the universal closure of the following formulas in the
language LT
1) Axiom of extensionality:
Axy(Az(zex<->zey) ->x=y)
2) Comprehension axiom schema: if F is a formula in which x is not
free then:
~F(V) -> ExAy(yex<->F)
is an axiom.
3) Completeness axioms for V
AxAy(xey&yeV -> xeV)
AxAy(x subset_of y & yeV -> xeV)
4) Reflection axiom schema: If z1,....,zn eV then
Ay(F(y,z1,...,zn)->yeV) -> Ex(xeV & Ay(yex<->F(y,z1,...,zn)))
is an axiom.
5) Axiom of regularity:
Ax( Ey(yex) -> Ey(yex & y disjoint x))
/
I would think that this theory do have proper classes as members of
other classes. for instance take F<->Vey
then x would be the class of all classes containing V as a member in
them, lets say it is V2, then we can have F<->V2ey
and we get V3 , etc..
Zuhair
.
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