Expressing cardinality quantifiers in monadic second-order logic over chains

We study the extension of monadic second-order logic of order with cardinality quantifiers “there exists infinitely many sets” and “there exists uncountably many sets”. On linear orders that require the addition of only countably many points to be complete, we show using the composition method that the second-order uncountability quantifier can be reduced to the first-order uncountability quantifier. In particular, this shows that the extension of monadic second-order logic with this quantifier has the same expressive power as monadic second-order logic over ordinals and over countable scattered linear orders. Using a Ramsey-like theorem of Shelah for dense linear orders we show how to eliminate the uncountability quantifier over the ordering of the rationals. Hence, ultimately, we give an elimination procedure that works over all countable linear orders. §