[FOM] Why would one prefer ZFC to ZC?

rgheck rgheck at brown.edu
Mon Feb 1 07:59:57 EST 2010

On 01/30/2010 07:38 PM, Jeremy Bem wrote:
> Tentatively, I believe in ZC and also hold a non-first-order belief
> that the countable union of sets is a set.  That is why I believe that
> these are sets, and indirectly why I believe that ZC is consistent.
This needs more careful formulation. We already have the full unions 
axioms in ZC, in the form:
     \forall x \exists y \forall z [ z \in y \equiv \exists w ( w \in y 
\wedge z \in w ) ]
Countable unions are included. So what you want is something quite 
different. The countable collection of sets whose union you want to take 
is not something we know antecedently is a set, viz: V_\omega, V_{\omega 
+ 1}, .... So the obstacle is not really the ability to take countable 
unions. It is the ability to form a set from an arbitrary countable 
collection. And that, to me, seems the right way to think of it. You 
would not want to formulate your axiom (scheme, it would have to be) of 
countable unions in such a way that you might have a union where you do 
not even have a set. (In fact, this will not happen, since of course we 
have the singletons of V_{\omega + n}. But that is an odd reason it will 
not happen.)

The point, then, is that the existence of the set of all the V_{\omega + 
n} is what is fundamentally at issue, and the intuition that is driving 
the assumption that this set exists is clearly of a piece with 
replacement. Replacement is just what you get if you generalize the same 
intuition beyond \omega.

I should perhaps say that I do not make these points as a fan of 
replacement. But I don't myself think ZC is a stable view. I think you 
either go up or you go down, and I'd rather go down. Most days.

> Presumably ZFC advocates (who are not pure formalists) hold a
> similarly external belief that "union over all ordinals" is a
> legitimate operation (albeit not one that returns a set).
I think it's profoundly controversial whether a "union over all 
ordinals" makes *any* kind of sense.


Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University

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