Stationary sets and infinitary logic

Abstract

Let K^0_lambda be the class of structures < lambda,<,A>, where A subseteq lambda is disjoint from a club, and let K^1_lambda be the class of structures < lambda,<,A>, where A subseteq lambda contains a club. We prove that if lambda = lambda^{< kappa} is regular, then no sentence of L_{lambda^+ kappa} separates K^0_lambda and K^1_lambda. On the other hand, we prove that if lambda = mu^+, mu = mu^{< mu}, and a forcing axiom holds (and aleph_1^L= aleph_1 if mu = aleph_0), then there is a sentence of L_{lambda lambda} which separates K^0_lambda and K^1_lambda .