[FOM] Definable Sets in ZFC

[Ali Enayat]

This is a sequel to my earlier posting on the same topic, and is
prompted by an e-mail from Jim Schmerl, who has kindly brought my
attention to the fact that in my previous reply to the questions posed
by Florian Rabe, I interpreted the questions in a manner that - prima
facie - do not coincide with Rabe's formulation.

More specifically, my posting concerned the collection D(M) consisting
of elements of a model M of ZF that are definable in M by some
parameter-free formula, whereas Rabe's questions can be taken to refer
to the subset d(M) of M consisting of elements m of M that are
"ZFC-definable", i.e., for some parameter-free formula F(x) of set
theory, F(m) holds in M, and ZFC proves "there is a unique x such that
F(x) holds".

However, as pointed out by Schmerl, one can use a trick to show that
d(M) = D(M), since given any formula F(x) of set theory, there is
another formula F*(x) with the property that F*(x) is a ZFC-definition
and for any model M, if F(x) defines m in M, then F*(x) also defines m
in M. For example, one can choose F*(x) to say:"if there is a unique y
such that F(y), then F(x); and otherwise x = 0".

Regards,

Ali Enayat