```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 ofZF) is the following:Language: FOL(=,e)Define: set(x) <-> Exist y. x e yAxioms: ID axioms +(1) Unique Construction: if P is a formula in which y occur free but xdo not, thenall closures of (Exist! x. for all y. y e x <-> set(y) & P) areaxioms.(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=nU(y) is the class union of y, defined in the customary manner./So simply MK is about unique construction of accessible sets, andproper classes of those.Zuhair
```