[FOM] countable saturation

[Ben Crowell]

In hopes of figuring out the paper by Kanovei and Shelah, "A definable
nonstandard model of the reals," http://shelah.logic.at/files/825.pdf ,
I'm studying the first two chapters of Model Theory by Chang and Keisler.
Regarding the term "countably saturated," there seems to be something I'm
not understanding, or else maybe K&S are using the term differently from
C&K. C&K say (slightly paraphrased to get it into ascii):
  A model U is said to be w-saturated iff for every finite subset Y of A,
  every set of formulas Gamma(x) of L_Y consistent with Th(U_Y) is
  realized in U_Y. A model is said to be countably saturated iff it is
  countable and w-saturated.
The Wikipedia article http://en.wikipedia.org/wiki/Saturated_model seems
to be describing a generalization of the concept, where instead of A
countable and Y finite, we can have bigger cardinalities, but still Y
lower in cardinality than A.

If I'm understanding this correctly, they're saying that they want the
richest possible set of types to exist, and the restriction to a finite
set of constants Y is to ensure that we don't have sets of formulas like
{x!=a, x!=b, x!=c, ...} that simply rule out all possible values for
x. What they do want to exist are ideal elements defined by sets of
formulas like {x>0, x>S(0), x>S(S(0)), ...}

What's throwing me for a loop is that K&S say in the main theorem on the
first page of the paper, "There exists a definable, countably saturated
extension *R of the reals R..." But C&K's definition of countably
saturated says the model has to be countable, whereas the hyperreals
K&S are constructing would be an uncountable set. Am I completely on
the wrong track here? Does "countable model" not mean what I think it
means?