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Topic: A compact reformulation of MK-Foundation-Choice.
Replies: 2   Last Post: Mar 23, 2013 2:31 PM

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Charlie-Boo

Posts: 1,585
Registered: 2/27/06
Re: A compact reformulation of MK-Foundation-Choice.
Posted: Mar 22, 2013 5:25 PM
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On Mar 19, 12:41 pm, Zuhair <zaljo...@gmail.com> wrote:
> This is also another reformulation of MK.
>
> Define: Set(x) iff Exist y. x in y
>
> Axioms:
>
> Unique construction: if phi is a formula in which x is not free, then:
> (Exist!x for all y ( y in x iff Set(y) & phi)) is an axiom.
>
> Pairing: (For all y (y in x -> y=a or y=b)) -> Set(x)
>
> Size limitation: Set(x) & |y| =< |H(TC(x))|  -> Set(y)
>
> /
>
> Definitions:
> TC(x)={y| for all t (x subclass_of t & t is transitive -> y in t)}
> t is transitive iff for all m,n (m in n & n in t -> m in t)
> H(x)= {y| for all z(z in TC({y}) -> |z| =< |x|)}
> |z| =< |x| iff Exist f (f:z -->x & f is injective).
> /
>
> This is a more compact presentation of MK-Foundation-Choice. Which

of
> course can interpret the whole MK and of course can construct a
model
> of ZFC thus proving its consistency.

But do you know that your system is consistent? Or if it has a
different set of theorems than ZFC (otherwise you are saying that if
ZFC is consistent then ZFC is consistent)?

C-B

> Zuhair




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