# [FOM] Why would one prefer ZFC to ZC?

Jeremy Bem jeremy1 at gmail.com
Sat Jan 30 19:38:26 EST 2010

Thanks.  I would say that the inability of ZC to establish the
existence of V_{omega*n}, for n = 1, 2, …, merits discussion in
general.  These do not appear to be controversial as sets.

Tentatively, I believe in ZC and also hold a non-first-order belief
that the countable union of sets is a set.  That is why I believe that
these are sets, and indirectly why I believe that ZC is consistent.

Presumably ZFC advocates (who are not pure formalists) hold a
similarly external belief that "union over all ordinals" is a
legitimate operation (albeit not one that returns a set).

-Jeremy

On Fri, Jan 29, 2010 at 11:24 PM,  <T.Forster at dpmms.cam.ac.uk> wrote:
> You can take the axiom of infinity in at least three forms
>
> (i)   The Von Neumann omega exists
> (ii)  Thee is a Dedekind-infinite set
> (iii) V_\omega exists
>
>
> In ZF they are all equivalent. Without replacement they are all
> inequivalent. See e.g. Adrian Mathias' *slim models* paper in the JSL.
>
>       tf