Because of the compactness
of first order logic, every structure has a proper elementarily equivalent extension.
However, in the countably compact language La!IQ obtained from first order logic by
adding a new quantifier Q and interpreting Qx as “there are at least ω1x’s such
that…,” the situation is radically different. Indeed there are structures of countable
type which are maximal in the sense of having no proper Lω1Q-extensions, and the
class S of cardinals admitting such maximal structures is known to be large. Here it
is shown that there is a countable complete Lω1Q theory T having maximal
models of cardinality κ for each κ ≧ 21 which is in S. The problem of giving a
complete characterization of the maximal model spectra of Lω1Q theories T
remains open: what classes of cardinals have the form Sp (T) = : there is a
maximal model of T of cardinality for T a (complete, countable) Lω1Q
theory.
: there is a
maximal model of 
for 