# [FOM] ZFC, the Universe, and Replacement

rgheck rgheck at brown.edu
Mon Feb 1 19:00:46 EST 2010

Much of what Jeremy has been saying seems to focus on the question
whether the acceptance of ZFC depends upon one's being able in some
sense to characterize the universe, V, of sets. But this thought can
be questioned. as it was by George Boolos in such papers as "On
Second-order Logic", "The Iterative Conception of Set", "Iteration
Again", and especially "To Be Is To Be a Value of a Variable" and
"Nominalist Platonism". The issue is also discussed in detail in Richard
Cartwright's paper "Speaking of Everything", and there are related
model-theoretic discussions in papers by Agustin Rayo, Gabriel Uzquiano,
Van McGee, and Timothy Williamson, among others. (Sorry if I've left
anyone out!)

In short, the response is that we should not suppose (a) that the
motivation for ZFC somehow depends upon a characterization of the
"totality" of sets and (b) we absolutely should not suppose that it
depends upon the presumption that the "totality" so characterized (the
"domain" of the theory) itself has to *be* a set. Jermey's emphasis on
the claim that we have no good reason to believe in the existence of a
"model" of ZFC, where this would be a *set* with certain properties,
suggests rather strongly that the advantages he claims for ZC rest upon
(a) and (b) or claims to the same effect.

It is an old idea, going back to the very origins of set-theory, that
neither of these should be accepted, though Cantor's own reasons for
rejecting (a) and (b) seem to have been a little peculiar, or at least
to have had some peculiar elements. As regards (a), the thought is that
a purely generative conception, as developed particularly in Scott's
axiomatization of stage theory', is adequate to motivate ZFC. As
regards (b), the thought is that this is just wrong, for reasons
Cartwright expresses particularly forcefully.

There are ways of developing some of these ideas, via notions like that
of "indefinite extensibility", that can lead to a somewhat contructivist
way of thinking, but there are plenty of people who think of the
set-theoretic universe as being in some sense "open-ended" who do not
have particularly constructivist leanings. Nor, I should add, am I
actually saying that one has to think of the set theoretic universe as
open-ended to reject (a) and (b). But it's a natural direction to take.

Somewhat ironically, later in his life Boolos expressed doubts about the
extent to which a "generative" motivation could be given for
replacement. He suggested, in "Iteration Again", I believe, that the
intutions that drive replacement might better be understood as based
upon a conception of set bound up with limitation of size' (a very
Cantorian idea). And he came to believe, toward the end of his life,
that maybe replacement wasn't very well motivated at all. But his
reasons were quite different and were ultimately based upon little more
than revulsion at the very idea that there might be objects as large as
those ZFC churns out. To quote:

...[J]ust what exactly is the matter with saying ZFC isn't correct
because it tells us that there are [lots] of objects and there aren't
that many objects? ...To be sure, one who says this may be asked how he
knows there aren't. But the reply, "Get serious. Of course there aren't
that many things in existence. I can't *prove* that there aren't, of
course, any more than I can *prove* that therea aren't any spirits shyly
but eagerly waiting to make themselves apparent when the Zeitgeist is
finally ready to acknowledge the possibility of their existence. But
there aren't any such spirits and there aren't as many things around as
[that]. You know that perfectly well, and you also know that any theory
that tells you otherwise is at best goofball."---that reply, although it
does not *offer reasons* for thinking that there are fewer than [that
many] objects in existence, would not seem to manifest any illusions
that could be called metaphysical realist. ["Must We Believe in Set
Theory", in *Logic, Logic, and Logic*, p. 145]

The remark about metaphysical realism isn't to the present point, but
one gets the idea.

For those who are wondering, the specific number Boolos has in mind is
the least ordinal \lambda for which \lamba = \aleph_\lambda. ZFC of
course entails the existence of such an ordinal.

Richard

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Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University