[FOM] Why would one prefer ZFC to ZC?

[Roger Bishop Jones]

On Tuesday 26 Jan 2010 Jeremy Bem wrote:

> 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".

Your argument rests on the supposition of an objective distinction between 
what is a set and what is a class.
However, this set/class distinction, though frequently spoken of as if 
objective, can only be sustained as a distinction relative to some given  
conception of set.
Relative to any definite transitive collection of well-founded sets, the 
classes are determined as the subcollections of the whole.
However, the sets and classes together, yield another definite conception of 
well-founded set, relative to which another collection of classes is 
determined.

You argue that V(omega+omega) should be taken as the universe of sets, i.e. 
the CLASS of sets and your reason for preferring this choice is that it 
"appears to be a set"!  Presumably, relative to the conception of set which 
you are intent on discarding.

The argument against there being any objective conception of V, the collection 
of all well-founded sets, has been discussed earlier on FOM in the thread:

	V does not exist

Which may be found in the archive at:

http://cs.nyu.edu/pipermail/fom/2005-October/009144.html

Roger Jones
-- 
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