[FOM] Question on the Axiom of Foundation/Regularity

Todd Eisworth eisworth at math.ohiou.edu
Tue Sep 25 11:06:26 EDT 2007

Other than the post by Jan Pax, one or two people gave me some suggestions
about where to look, but nothing turned up in those references.

The ``set-like'' assumption is certainly necessary if you want to use
transfinite recursion on the well-founded relation; since this is a pretty
gosh-darned important part of the development of set theory, my guess is
that the distinction between (*) and (**) (if there is one) is probably not
something that ever garnered much attention. 


> did you get any satisfactory replies to this?  (Ignore my 
>comment about a descending sequence of length or ordinal of On 
>- that was rubbish.)  I can see easily how to prove the 
>equivalence of we have DC -  but without DC i don't see how to 
>do it.  If you got any informative feedback i would interested 
>to see it.


> 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".

> So, are there models of ZF - Foundation lurking out there in the weeds 
> in which there are R and A for which (*) holds, and yet some instance 
> of (**) is false, or is the "set-like" assumption not really necessary 
> when working in ZF-foundation, and only assumed for convenience?

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