[FOM] Revisiting "limitation of size" axiom.
Zuhair Abdul Ghafoor Al-Johar
zaljohar at yahoo.com
Thu Sep 15 08:49:34 EDT 2016
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,
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:
More information about the FOM