Date: Nov 21, 2012 3:31 AM
Author: Zaljohar@gmail.com
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

axioms.

(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< x)

Def.) y is reachable by union <-> (Exist y. y < x & x =< U(y))

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.

Zuhair