[FOM] Question on the Axiom of Foundation/Regularity

Todd Eisworth eisworth at math.ohiou.edu
Mon Sep 17 15:49:58 EDT 2007

In class today, a question arose on the ways in which weakenings of ZF deal
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 Eisworth
Department of Mathematics
Ohio University

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