Fuhrken has shown that the
language Lω1Q, obtained from first order logic by adding a new quantifier Q and
interpreting QX as “there are at least ω1x’s such that …” is countably compact but
not fully compact. The countable compactness is not enough to yield strong
analogs of the upward Löwenheim-Skolem theorem, and the amalgamation
property. In fact, it is shown that for “most” cardinals κ, there are structures of
power κ with countable type that are maximal in the sense of having no
proper extensions with the same Lω1Q theory. From this the failure of the
amalgamation property is obtained. There is still the possibility that the
model theory of LκQ (with QX interpreted as “there are at least κx’s such
that …”) for κ > ω1, is more analogous to the model theory of first order
logic.
