### 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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