We investigate the extension of monadic second-order logic of order with cardinality quantifiers “there exists uncountably many sets such that …” and “there exists continuum many sets such that …”. We prove that over the class of countable linear orders the two quantifiers are equivalent and can be effectively and uniformly eliminated. Weaker or partial elimination results are obtained for certain wider classes of chains. In particular, we show that over the class of ordinals the uncountability quantifier can be effectively and uniformly eliminated. Our argument makes use of Shelah's composition method and Ramsey-like theorem for dense linear orders.
Vince Bárány. łukasz Kaiser. Alexander Rabinovich. "Expressing cardinality quantifiers in monadic second-order logic over chains." J. Symbolic Logic 76 (2) 603 - 619, June 2011. https://doi.org/10.2178/jsl/1305810766