Transitive closures between ZF minus and MK minus

[zuhair]

Hi all,

The transitive closure of x denoted as "TC(x)" is defined as the
minimal transitive set that has x as a subset of it.

In symbols:

y=TC(x) iff (y is transitive & x subset of y &
for all z ((z is transitive & x subset of z)
-> y subset of z)).

Now in ZF minus Regularity, we have the following Lemma:

For all x , for all y

y e TC(x) if and only if there exist a finite sequence
<x0,x1,x2,...,xn> were x0 e x and
xi+1 e xi for every i=0,1,2,...,n-1 and y=xn.

However is that lemma a theorem of MK minus Regularity also?

Zuhair

[Aatu Koskensilta]

zuhair <zalj...@gmail.com> writes:

> However is that lemma a theorem of MK minus Regularity also?

Every theorem of ZF minus regularity is also a theorem of MK minus
regularity.

--
Aatu Koskensilta (aatu.kos...@uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

[zuhair]

On Dec 29, 6:03 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:

> zuhair <zaljo...@gmail.com> writes:
> > However is that lemma a theorem of MK minus Regularity also?
>
> Every theorem of ZF minus regularity is also a theorem of MK minus
> regularity.
>
> --

> Aatu Koskensilta (aatu.koskensi...@uta.fi)

>
> "Wovon man nicht sprechan kann, dar ber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Ok, thanks a lot.

Zuhair