# [FOM] Hierarchy Set Theory: A Landscape of ZF.

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Wed Oct 26 07:36:58 EDT 2011

```HST is a class theory in first order logic with equality and membership

Ontology: a pure class theory, no Ur-elements, just sets and proper
classes.

Define (set): x is a set iff Exist y. x e y

Axioms:
1.Extensionality
2.Impredicative class comprehension
3.Ordinals: every accessible ordinal is a set.

Define (V_i):
V_0 = 0
V_i =P(V_i-1) for any successor set ordinal i
V_i =U(V_j) for all j<i for any limit set ordinal i

4.Hierarchy. For every set ordinal i: V_i is a set
and every subclass of V_i is a set.
/

Note: P and U symbolizes Power class and Union class
respectively in the standard manner.

Now this theory is at least Morse-Kelley. However
What i am interested in is building the model of ZF
in it, this is easily done, we define
V=U{V_i| i is accessible ordinal}
then define x is a ZF_set iff x e V
define the relations e* and =* which are just
e and = restricted to V. State all axioms
of ZF in terms of e* and =* over ZF_sets.
It is easy to see that all of those statements
hold in HST.

This theory shows ZF from an external angle, it shows
its model, and thus clarifies what ZF is all about,
the ordinary presentation of ZF is an internal view of
ZF, it shows the tools by which the model of ZF is built
but here one SEE the model clearly constructed by stages
indexed by accessible ordinals, impredicative class comprehension
plays a powerful role here, and its the most natural of all
comprehension schemes i saw thus far, Extensionality is only
inherent in the concept of set. So this axiomatic system is
not arbitrary at all, it is clearly motivated.

This theory can show ZF for what it is not for what is built by.

The idea of this theory is not original at all, it goes back to
Von Neumann's presentation  of the cummulative hierarchy which
is the model of ZF, it is pretty much a very old idea.
However it is nice to see it depicted in this particular manner.

Zuhair
```