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

rgheck rgheck at brown.edu
Mon Feb 1 07:42:49 EST 2010

On 01/30/2010 09:17 PM, Monroe Eskew wrote:
> On Fri, Jan 29, 2010 at 4:50 PM, Jeremy Bem<jeremy1 at gmail.com>  wrote:
>
>> How about countable union?
>>
> Now V{\omega+\omega} satisfies the Union axiom, and a fortiori an
> axiom of countable unions.  So I think you really mean countable
> replacement.  (If you have a formula that defines for each n in omega
> a unique object, then there is a set containing all those objects,
> e.g.  n |-->  V_omega+n.)  But this raises the question: Why not full
> replacement?  What's so special about omega in this respect?
>
>
This is of course the right question. And the standard justification of
replacement is something like: There's nothing special about omega; the
same intuition ought to go through for any ordinal you understand; but
that just does give you replacement, at least in the presence of choice.

Richard