The Axiom of Choice, the Löwenheim-Skolem Theorem and Borel models
The number of -equivalent nonisomorphic models for κ weakly compact
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 -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 Σ¹₁-definable over . By [SV] it is possible to have a generic extension where the possible numbers of equivalence classes...