[FOM] Question on the Axiom of Foundation/Regularity

Thomas Forster T.Forster at dpmms.cam.ac.uk
Tue Sep 25 22:52:30 EDT 2007

I'm wondering if Todd got any satisfactory replies to this?  (Ignore my
comment in an earlier post about a descending sequence of length of
ordinal of On - that was rubbish.)  I can see easily how to prove the 
equivalence if we have DC -  but without DC i don't see how to do it. If 
there was any informative feedback i would be interested to see it.
 Mon, 17 Sep 2007, Todd Eisworth wrote:
 > In class today, a question arose on the ways in which weakenings of ZF deal
 > with
 > well-founded relations on proper classes. Since this isn't the type of thing
 > I'm used to thinking about,
 >  I thought I would ask the FOM community.
 > In particular, suppose R and A are (proper) classes, with R a relation on A
 > that "linearly orders" A.
 > 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".
 > I know that if we are working in full ZF, then any instance of (**) is
 > provable from the statement (*).
 > In addition, if we know that R is set-like ({y in A: y R x} is a set for all
 > x in A), then ZF - Foundation will still get us (**). 
 > 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?
 > Best Wishes,
 > Todd
ome page: www.dpmms.cam.ac.uk/~tf; dpmms phone +44-1223-337981. 
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