[FOM] Question on the Axiom of Foundation/Regularity
Jan Pax
pax0 at seznam.cz
Mon Sep 17 23:46:05 EDT 2007
On: 17.9.2007 22:21:05 Todd Eisworth <eisworth at math.ohiou.edu> wrote:
> 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?
>
(*) and (**) are equivalent in ZF-foundation without further assumptions.
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