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

Jeremy Bem jeremy1 at gmail.com
Fri Jan 29 19:50:14 EST 2010

Monroe Eskew wrote:

>> (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.)

Meanwhile, V simply cannot be viewed as a model in the ordinary sense.

> 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.

I am speaking about V itself.  I am showing that the left hand side of
the analogy ZC : V_{omega+omega} :: ZFC : V is fundamentally better
behaved than the right.

>> (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.

Correct.

> What might it be?

For the moment, the argument above!

-Jeremy