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

Monroe Eskew meskew at math.uci.edu
Thu Jan 28 18:16:35 EST 2010

> (2) It should be possible to view the "universe" as a model in the
> ordinary sense of first-order logic;

The question is: Possible by what means?  It is not possible to prove
V{\omega+\omega} exists in ZC.  You may simply assume it without
accepting full replacement.  But why?  To me it seems whatever
intuition makes you think that V{\omega+\omega} exists would also make
you think replacement is true.  (Not in terms of logical necessity but
concepts.)

Likewise there are a variety of relatively mild assumptions, with some
intuitive backing, that give a natural transitive model of ZFC.  For
example inaccessibles can be seen as justified by some closure
principle.  These assumptions go beyond ZFC but likewise \omega*2 goes
beyond ZC.

> (3) It should be possible to prove beautiful, generally accepted
> mathematical results using the axiom system -- the more the better.

Then this should push you towards replacement, unless there is
something holding you back.  What might it be?