[FOM] Why would one prefer ZFC to ZC?

[T.Forster@dpmms.cam.ac.uk]

Because there is so much mathematics that can be done in ZFC but not 
Zermelo.


On Jan 27 2010, 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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