### Stationary Sets and Infinitary Logic

by Shelah and Vaananen. [ShVa:657]

J Symbolic Logic, 2000

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 .

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