### The number of $L_{\infty\kappa}$--equivalent nonisomorphic models for $\kappa$ weakly compact

by Shelah and Vaisanen. [ShVs:718]

Fundamenta Math, 2002

For a cardinal kappa and a model M of
cardinality kappa let No (M) denote the number
of non-isomorphic models of cardinality kappa which are
L_{infty kappa} --equivalent to M . In [Sh:133] Shelah
established that when kappa is a weakly compact cardinal and
mu
<= kappa is a nonzero cardinal, there exists a model M of cardinality
kappa with No (M)= mu . We
prove here that if kappa is a weakly compact cardinal, the
question of the possible values of No (M) for
models M of cardinality kappa is equivalent to the
question of the possible numbers of equivalence classes of
equivalence relations which are Sigma^1_1-definable over
V_kappa . In [ShVa:719] we prove that, consistent wise, the
possible numbers of equivalence classes of Sigma^1_1-equivalence
relations can be completely controlled under the singular cardinal
hypothesis. These results settle the problem of the possible values
of No (M) for models of weakly compact
cardinality, provided that the singular cardinal hypothesis holds.

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