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

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Fri Mar 29 04:33:45 EDT 2019

Dear Thomas Klimpel,

My notes will be inserted among yours, see below:

On Wed, 27 Mar 2019 13:21:55 +0100  Thomas Klimpel <jacques.gentzen at gmail.com> wrote:

I would say it is simply not sufficiently different from ZFC. Remember

that reference to ZFC implies the possibility to add some inaccessible

cardinals if necessary. For example, HoTT with propositional

truncation and choice on the level of sets needs two inaccessibles,

and this is not considered to be a big deal. But HoTT itself is so

different from ZFC in its basic approach that it is worthwhile to use

it as an alternative foundation
.......

The reason for electing it as a foundation theory made here is
because of its connection to Muller's arguments! This theory would

generally satisfy those requirements, while ZFC and its extensions do not!
.......

Muller's article also uses an explicit constant for the class of all
sets, but he uses "V" instead of "W". Assuming you based your theory

on the description from Muller's article, why did you replace "V" by

"W"? How is your axiomatic theory related to the axiomatic theory

presented in Muller's article? Is it the same? Or is it more powerful?

....

V was changed to W, because V is used in this theory to symbolize the class of
all *sets* [i.e., elements of classes] which is definable in the pure language of set theory, much as its done in Morse-Kelley set theory, while V of Muller's is about a sub-world of sets, and in this theory this corresponds to W, which is not definable in the pure language of set theory.

Regarding its power relative to Muller's system. This system is more powerful!
However the extra abundance didn't result from a try to capture Categories

here, the resulting abundance is about sets per se, so it doesn't generally
oppose Muller's first objection of "Superabundancy"
.......
> 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.

I don't remember having seen this axiom scheme before, but I may be

wrong. Is this your invention, or is this basically equivalent to the

"Axiom of Completeness (Compl). The class of all sets V is complete,

i.e. all classes contained or included in sets are sets."

from Muller's article (or some axiom scheme from another reference)?
......

I personally didn't see this axiom mentioned before, but it's likely the

chance that it was mentioned. However the axiom scheme doesn't by itself imply
completeness over W, but it would interpret it over the class H_W of all

hereditarily [elements of W] sets, and over that realm it would interpret all
axioms of Ackermann's set theory [thus interpreting ZFC]. All axioms of Muller's theory would be interpreted over the whole realm of finite iterative powers of H_W,

However this theory is stronger than ZFC. I think it can be interpreted in
ZFC+ 0^\sharp.........

Best Regards,
Zuhair