[FOM] Why would one prefer ZFC to ZC?

[Jeremy Bem]

I am not overlooking basic considerations related to Godel's
incompleteness theorem, nor misusing the set/class distinction, nor
necessarily "using replacement" when I assert that V_{omega+omega}
"is" a set whereas V "is" a proper class.  There are many ways in
which this distinction can be justified.

For example, I am not aware of any context or system in which V (:=
the union over all ordinals alpha of V_alpha) can be understood as a
set.  In contrast, it is quite ordinary and reasonable to believe that
V_{omega+omega} is a set, whether or not one accepts the entire
replacement axiom schema.

Formally, one might work in a system stronger than any under
consideration, such as Morse-Kelley set theory.  One could first
establish the standard result of first-order logic that if a theory
has a (set-sized) model, then it is consistent.  One could then
*apply* this result to V_{omega+omega} to show that ZC is consistent.
However, this would not work for (V, ZFC) without at least some
modification, because of the set/class issue that would arise.

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We could thus enumerate the following desiderata for a foundational
axiom system:

(1) There should exist an associated "universe" construction, such
that a Platonist might claim to simply perceive that the axioms are
true when interpreted in the universe;

(2) It should be possible to view the "universe" as a model in the
ordinary sense of first-order logic;

(3) It should be possible to prove beautiful, generally accepted
mathematical results using the axiom system -- the more the better.

Do we agree upon these?  Moreover, isn't (3) a matter of degree,
whereas (2) is a matter of principle?  As mathematicians and
philosophers, shouldn't we care more about the principle?

Jeremy