On equivalence relations second order definable over $H(\kappa)$

by Shelah and Vaisanen. [ShVs:719]
Fundamenta Math, 2002
Let kappa be an uncountable regular cardinal. Call an equivalence relation on functions from kappa into 2 Sigma_1^1-definable over H(kappa) if there is a first order sentence phi and a parameter R subseteq H(kappa) such that functions f,g in {}^kappa 2 are equivalent iff for some h in {}^kappa 2, the structure (H(kappa), in,R,f,g,h) satisfies phi, where in, R, f, g, and h are interpretations of the symbols appearing in phi . All the values mu, 1 <= mu <= kappa^+ or mu =2^kappa, are possible numbers of equivalence classes for such a Sigma_1^1-equivalence relation. Additionally, the possibilities are closed under unions of <= kappa-many cardinals and products of < kappa-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 Sigma_1^1-equivalence relations might consistent wise be exactly those cardinals which are in a prearranged set, provided that the singular cardinal hypothesis holds and that some necessary conditions are fulfilled.

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