# [FOM] Revisiting "limitation of size" axiom.

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Thu Sep 15 08:49:34 EDT 2016

```Dear Sirs,

The following axiom when added to axioms of Extensionality, Class comprehension
schema and axiom of infinity, all of Morse-Kelley set theory, then it would prove:
Pairing, Union, Power, Separation, Replacement and Global choice, all in one go.

Limitation of size: for all y [y in V <-> Exist x (y =< x ^ Ux < V)]

In English: a set is what is subnumerous to a class whose union is strictly
subnumerous to the class of all sets.

Where V is the class of all sets; =< denotes 'is subnumerous to' and < denotes "is
strictly subnumerou to" defined as:
x =< y <-> x subset_of y or Exist f(f:x-->y ^ f is injective)
x < y <-> x =< y ^ ~ y =< x.
U is the class union operator defined in the customary manner.

A modification of this axiom would prove all of the above axioms except Global choice,
that is:

Limitation of size: for all y,x [y =< x ^ Ux in V -> y in V ^ Uy in V]

I personally figured out those axioms during trying to develop a Mereological foundation of set theory, and in some sense the above axioms were motivated by it.

For full proof see: