[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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