[FOM] The Cumulative Hereditary Hierarchy H.

[Zuhair Abdul Ghafoor Al-Johar]

Dear FOMers,

Define the Cumulative Hereditary Hierarchy H as the union of
all the following stages:

H_0=0
H_i+1=H(H_i) for any Von Neumann ordinal i
H_i= U(H_j) j<i, for any limit Von Neumann ordinal i

where H(x) is the set of all sets hereditarily subnumerous to x.

Questions:

1) Can ZF alone (i.e. without Choice) prove the existence of H_i for
each ordinal i ?

2) Is the cumulative hierarchy V of Von Neumann's a subclass of H ?

3) If 2) is true, then can there exist a set in H that is not equinumerous
to any set in V.

Happy new year!

Zuhair