Hi Tim,
What you refer to with transfinite numbers and different sizes of
infinities is exactly what happens when the semantics are dispensed
with.
Regarding "THE intersection
of an infinite set of models"... think about it: There is no singular "largest
model" in an infinite set. There is no one "intersection". However, I can
answer one aspect of that question: Any intersection of two (or more) models
is also a model. In an infinite set, there will be infinite
intersections.
Judith