On equivalence relations Σ1-definable over H(κ)

Let κ be an uncountable regular cardinal. Call an equivalence relation on functions from κ into 2 Σ11-definable over H(κ) if there is a first order sentence φ and a parameter R ⊆ H(κ) such that functions f, g ∈ κ2 are equivalent iff for some h ∈ κ2, the structure 〈H(κ),∈, R, f, g, h〉 satisfies φ, where ∈, R, f , g, and h are interpretations of the symbols appearing in φ. All the values μ, 1 ≤ μ ≤ κ or μ = 2κ, are possible numbers of equivalence classes for such a Σ11equivalence relation. Additionally, the possibilities are closed under unions of ≤ κ-many cardinals and products of < κ-many cardinals. We prove that, consistent wise, these are the only restrictions under the singular cardinal hypothesis. The result is that the possible numbers of equivalence classes of Σ11-equivalence relations might consistent wise be exactly those cardinals which are in a prearranged set, provided that the singular cardinal hypothesis holds and that the following necessary conditions are fulfilled: the prearranged set contains all the nonzero cardinals in κ ∪ {κ, κ, 2κ} and it is closed under unions of ≤ κ-many cardinals and products of < κ-many cardinals. The result is applied in [SV] to get a complete solution of the problem of the possible numbers of strongly equivalent non-isomorphic models of weakly compact cardinality. 1 ∗Research supported by the United States-Israel Binational Science Foundation. Publication 719. 1991 Mathematics Subject Classification: primary 03C55; secondary 03E35, 03C75.