[FOM] A question about Ackermann's set theory.

[Zuhair Abdul Ghafoor Al-Johar]

Dear Sirs,

The reflection schema in Ackermann's set theory states that every formula
with set parameters that do not use the set-hood predicate symbol and that
only hold for sets can define a set.

Now we can define standard von Neumann ordinal in the following manner:

x is a von Neumann ordinal iff x is transitive and every element of x is
transitive and every non empty subclass of x has a disjoint element of it.

x is a standard von Neumann ordinal iff x is a von Neumann ordinal and
(for all y. y is a von Neumann ordinal and y is a subclass of x and not y=x
-> y e x)

Now define accessible standard von Neumann ordinal as a standard von
Neumann ordinal where all its subclasses are not uncountable regular limit cardinals.

Now the predicate "accessible standard von Neumann ordinal" fulfills all
criteria stipulated in the reflection scheme of Ackermann's, so the class
{x: x is accessible standard von Neumann ordinal} is a set and obviously
it must be inaccessible.

But this would mean that Ackermann's set theory can tell us something
about sets that ZF cannot, the above assertion is not provable in ZF.

Doesn't that contradict known results about equivalence of ZF and
Ackermann's over the realm of sets?

Best Regards

Zuhair