[FOM] Why would one prefer ZFC to ZC?

[Jeremy Bem]

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