Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Paraphrasing MK
Posted:
Nov 21, 2012 3:28 AM


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 & 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.
Zuhair



