[FOM] Why would one prefer ZFC to ZC?

[Arnon Avron]

How do you know that "V_{omega+omega}" exists, and that
it is a set, without accepting replacement?

Arnon Avron

On Tue, Jan 26, 2010 at 02:09:27PM -0800, Jeremy Bem wrote:
> Hi!
> 
> In a recent thread entitled "Mac Lane set theory", I tried to explain
> why I tentatively prefer ZC to ZFC (as a foundation for mathematics).
> To summarize that argument: whatever role the "Von Neumann universe"
> plays in justifying ZFC, can be played for ZC by "V_{omega+omega}".
> But whereas the former construction is said to yield a proper class,
> the latter appears to be a set -- making it a model in the ordinary
> sense of first-order logic.
> 
> Arguably, the existence of such a construction makes ZC qualitatively
> more justified than ZFC.  As an informal consistency proof, it is an
> application of ordinary model theory, rather than a unique argument
> involving an exotic "union over all ordinals" and a class-sized
> "model".
> 
> I don't claim that this argument is overwhelmingly compelling, but it
> is an argument.  Why would someone prefer ZFC?
> 
> -Jeremy
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