[FOM] Why would one prefer ZFC to ZC?

[Jeremy Bem]

Monroe, I'm having a hard time following you.

On Sat, Jan 30, 2010 at 6:17 PM, Monroe Eskew <meskew at math.uci.edu> wrote:
> Now V{\omega+\omega} satisfies the Union axiom, and a fortiori an
> axiom of countable unions.

What does this mean?

> So I think you really mean countable replacement.

I meant what I said; that one reason someone might believe that
V_{omega+omega} is a set, is that they believe in ZC, and also believe
that the countable union of sets is a set.

> But this raises the question: Why not full
> replacement?  What's so special about omega in this respect?

Omega is special in many ways, but I'll let you clarify first.

>> 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.
>
> I don't disagree that a set is better behaved than a class, but I
> don't think you're making a fair analogy.  You are saying that ZC is
> good because only a slight extension of it implies the existence of a
> natural model of it.

No, I'm saying that the "standard foundations of mathematics" seem to
involve a construction called the Von Neumann universe, or V, in some
sort of justificatory role -- and that if one were to replace ZFC by
ZC, that role could be played by V_{omega+omega} instead.  Is that
unfair?  And the latter is better behaved, as you've agreed.

V is defined as the union, over all ordinals alpha, of V_alpha.  Does
V not play a role in your understanding of, and affection for, ZFC?
That would be interesting.

> Furthermore since the slight extension that of ZC you have in mind is
> merely a fragment of the replacement schema, it seems ironic (if not
> inconsistent) to use this to argue against replacement.

How is closure under countable unions a fragment of the replacement
schema?  It doesn't even mention first-order formulas (and isn't
first-order).  I'm not even sure whether all models of ZFC are closed
under countable unions.

-Jeremy