[FOM] Why not this theory be the foundational theory of mathematics?

[Zuhair Abdul Ghafoor Al-Johar]

All axioms of class theory and set theory can be interpreted in a first order
theory (with equality) plus primitives of set membership "\in", and "W" standing
for some fixed set. 

Define: set(y) <-> \exists x (y \in x)

We only need two axioms:

1. Class Comprehension schema: if \phi is a formula in which x is not free, then
all closures of: (\exists x forall y (y \in x <-> \phi & set(y))) are axioms.

In English: for every formula \phi there exists a class of all and only sets that
satisfy \phi 

2. Set Comprehension schema: if \phi is a formula in the pure language of set
theory [i.e., doesn't use the symbol "W"], then: 

x_1,..,x_n \in W -> [\forall y (\phi -> y \subset W) -> \forall y(\phi -> y \in W)],

is an axiom.

In English: any pure set theoretic formula from parameters in W, that only
holds of subsets of W; also only holds of elements of W.

This theory can interpret ZF-Foundation-Extensionality, over the realm of
hereditarily elements of W sets. And thus can provide a full interpretation of ZFC.


This theory is much more powerful than ZFC.

The axiomatics for this theory seem to be very natural to me. Its astoundingly simple,

yet its very strong. 

This theory also fulfill all of F.A. Muller's criteria for a founding theory of mathematics!

for Muller's criteria see: 

http://philsci-archive.pitt.edu/1372/1/SetClassCat.PDF


Why wan't such a simply presented natural theory qualify as the foundational theory of mathematics? 

Zuhair