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Re: Paraphrasing MK
Posted:
Nov 21, 2012 10:34 AM


On Nov 21, 12:31 am, Zuhair <zaljo...@gmail.com> wrote: > 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 P and NP(1) respectively [MIN. c, ch, K]. In particular(.) the NP complete(?ch) languages have been studied intensively and virtually hundreds of natural NP complete (replete) problems have been found in many different areas oMus@ov(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



