Date: Nov 21, 2012 3:25 AM
Subject: Paraphrasing MK

Another way to present MK (which of course prove the consistency of
ZF) is the following:

Language: FOL(=,e)

Define: set(x) <-> Exist y. x e y

Axioms: ID axioms +

(1) Unique Construction: if P is a formula in which y occur free but x
do not, then
all closures of (Exist! x. for all y. y e x <-> set(y) & P) are

(2) Size: Accessible(x) -> set(x)

Where Accessible(x) is defined as:

Accessible(x) <-> (Exist maximally two m. m e x) OR
~ Exist y:
y subset of x &
y is uncountable &
y is a limit cardinal &
y is not reachable by union.

Def.) y is a limit cardinal <-> (for all y. y < x -> Exist z. y < z <
Def.) y is reachable by union <-> (Exist y. y < x & U(y) =< x)

The relation < is "strict subnumerousity" defined in the usual manner.
The relation =< is subnumerousity defined in the usual manner.
"subset of" and "uncountable" also defined in the usual manner.
"Exist maximally two m. phi(m)" is defined as
Exist m,n for all y. phi(y) -> y=m or y=n
U(y) is the class union of y, defined in the customary manner.

So simply MK is about unique construction of accessible sets, and
proper classes of those.