The number of L∞κ-equivalent non-isomorphic models for κ weakly compact
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چکیده
For a cardinal κ and a model M of cardinality κ let No(M) denote the number of non-isomorphic models of cardinality κ which are L∞,κequivalent 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 Σ1-definable over Vκ. By [SV] it is possible to have a generic extension, where the possible numbers of equivalence classes of Σ1-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. 1
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