[FOM] Question on the Axiom of Foundation/Regularity

Thomas Forster T.Forster at dpmms.cam.ac.uk
Tue Sep 18 18:40:24 EDT 2007


would you care to tell us why?  I can see how to do it in ZF - foundation 
+ DC, but i blush to admit that i can't see how to do it without DC (and
i'm supposed to know about set theory without foundation!).  Why shouldn't 
A have a descending R-chain of order type Kappa* where Kappa is the order 
type of of On, and is regular?  How does one get a subset with no 
R-minimal element out of that?  (perhaps by `ZF' you meant `ZFC'...?)

      puzzled of pmms

On Tue, 18 Sep 2007, Jan Pax wrote:

> >  Let (*) be the statement
> >  
> >  "every non-empty subset of A has an R-minimal element"
> >  
> >  and let (**) be the scheme corresponding to (the informal)
> >  
> >  "every non-empty subclass of A has an R-minimal element".
> >  
> (*) and (**) are equivalent in ZF-foundation without further assumptions.
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