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791
Registered:
9/1/10
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Re: Paraphrasing MK
Posted:
Nov 21, 2012 10:34 AM
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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
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