[FOM] Why would one prefer ZFC to ZC?

[rgheck]

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