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

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Sat Mar 23 14:24:38 EDT 2019

Dear Sirs,

The following is a link to clearer exposition of this theory:

Best regards,
Zuhair

On Saturday, March 23, 2019 7:08 PM, "fom-request at cs.nyu.edu" <fom-request at cs.nyu.edu> wrote:

Date: Thu, 21 Mar 2019 20:12:45 +0000 (UTC)

From: Zuhair Abdul Ghafoor Al-Johar <zaljohar at yahoo.com>

To: Foundations of Mathematics <fom at cs.nyu.edu>

Subject: [FOM] Why not this theory be the foundational theory of

mathematics?

Message-ID: <957965873.9400098.1553199165360 at mail.yahoo.com>

Content-Type: text/plain; charset=UTF-8

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