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

Monroe Eskew meskew at math.uci.edu
Tue Jan 26 23:20:59 EST 2010

You need the axiom of replacement to prove the existence of
V_{\omega*2}.  So your argument requires the acceptance of ZFC in the
background.

It's similar to saying the following:  If \kappa is the least
inaccessible, V{\kappa} is a model of ZFC.  Hence there is a set model
of ZFC without inaccessibles.  Thus we don't need to believe in
inaccessibles.

Godel's incompleteness theorem sheds a lot of light here.

Monroe

On Tue, Jan 26, 2010 at 2:09 PM, Jeremy Bem <jeremy1 at gmail.com> wrote:
> To summarize that argument: whatever role the "Von Neumann universe"
> plays in justifying ZFC, can be played for ZC by "V_{omega+omega}".
> But whereas the former construction is said to yield a proper class,
> the latter appears to be a set -- making it a model in the ordinary
> sense of first-order logic.