# [FOM] Con(ZF) proved from three basic themes

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Wed Dec 14 04:07:18 EST 2011

```Dear Sir(s),

A weak version of this theory but enough to prove Con(ZF)
is the following:

EXPOSITION: T' is a theory in FOL(=,e) with the following axioms:

Def.) elm(x):= [y](x e y)
Def.) trs(x):= (y)(y e x -> y c x)

Axioms:
I. Construction: if phi is a formula in which x is not free, then
([!x](y)(y e x <-> elm(y) & phi)) is an axiom.

II. Relations: (a)(b)(x)((y)(y e x -> y=a ? y=b) -> elm(x))

Def.) x < y:= [f](f:x-->y & f is injection)
Def.) U(x)=y <-> (z)(z e y <-> [m](m e x & z e m))

III. Size axioms:

(x)(y)(elm(x) & y < x -> elm(y))
(x)(y)(elm(x) & trs(y) & (z)(z e y -> z < U(x)) -> elm(y))

/

Zuhair Al-Johar

At Mon, 12 Dec 2011 03:12:42 -0800 (PST) Zuhair Abdul Ghafoor Al-Johar <zaljohar at yahoo.com> wrote:
>
> Dear FoMers,
>
>  The following theory of mine interpret ZF under
> the realm of well founded elements of it, it also
> proves the consistency of ZF by defining its model.
>
> ONTOLOGY: every object this theory speaks about is
> termed as a "set", however there are two kinds of sets,
> those that are elements of sets, those are termed as
> "elements", and those that are not elements of any
> set, those are termed as "proper sets". This theory
> do not contain objects that are elements of sets but
> are themselves not sets (i.e. proper elements), so
> all elements here are sets, but not all sets are elements.
> So the term "set" here is used instead of the term "class"
> used in Morse-Kelley's; while the term "element" here is
> used instead of the term "set" used in Morse-Kelley's.
>
>
> Key to notations is found at: http://zaljohar.tripod.com/logic.txt
>
> Best Regards

```