The number of L∞κ-equivalent nonisomorphic models for κ weakly compact

Abstract

For a cardinal κ and a model M of cardinality κ let No(M) denote the number of nonisomorphic models of cardinality κ which are L∞,κ-equivalent to M. We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are ∑11-definable over Vκ. By [SV] it is possible to have a generic extension where the possible numbers of equivalence classes of ∑11-equivalence relations are in a prearranged set. Together these results settle the problem of the possible values of No(M) for models of weakly compact cardinality.