The pair $(\aleph_n,\aleph_0)$ may fail $\aleph_0$--compactness

by Shelah. [Sh:604]
Proc LC'2001, 2005
Let P be a distinguished unary predicate and K= {M: M a model of cardinality aleph_n with P^M of cardinality aleph_0} . We prove that consistently for n=4, for some countable first order theory T we have: T has no model in K whereas every finite subset of T has a model in K . We then show how we prove it also for n=2, too.

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